text:
\section{Introduction}
Matrix recovery problems of the form
\begin{equation}
\min_{X} f({\bm \sigma}(X))) + \|{\mathcal{A}} X - b\|^2,
\label{eq:general}
\end{equation}
where ${\mathcal{A}}$ is a linear operator and ${\bm \sigma}(X) = (\sigma_1(X),\sigma_2(X),...)$ are the singular values of $X$,
are frequently occurring in computer vision. Applications range from high level 3D reconstruction problems to low level pixel manipulations \cite{tomasi-kanade-ijcv-1992,bregler-etal-cvpr-2000,yan-pollefeys-pami-2008,garg-etal-cvpr-2013,basri-etal-ijcv-2007,garg-etal-ijcv-2013,wang-etal-2012,canyi-etal-cvpr-2014,oh-etal-pami-2016,hu-etal-pami-2013,gu-2016}.
In structure from motion the most common approaches enforce a given low rank $r$ without additionally penalizing non-zero singular values \cite{tomasi-kanade-ijcv-1992,bregler-etal-cvpr-2000,hong-fitzgibbon-cvpr-2015}.
(This can be viewed as a special case of \eqref{eq:general} by letting $f$ assign zero if fewer than $r$ singular values are non-zero and infinity otherwise.)

Since the rank of a matrix $X$ is bounded by to the number of columns/rows in a bilinear parameterization $X = B C^T$, the resulting optimization problem can be written
\begin{equation}
\min_{B,C} \|{\mathcal{A}}(BC^T)-b\|^2.
\end{equation}
This gives a smooth objective function and can therefore be optimized using 2nd order methods.
In structure from motion problems, where the main interest is the extraction of camera matrices from $B$ and 3D points from $C$, this is typically the preferred option \cite{buchanan-fitzgibbon-cvpr-2005}. In a series of recent papers Hong \textit{et al}.  showed that optimization with the VarPro algorithm is remarkably robust to local minima converging to accurate solutions  \cite{hong-fitzgibbon-cvpr-2015,hong-etal-eccv-2016,hong-etal-cvpr-2017}.
In \cite{hong-zach-cvpr-2018} they further showed how uncalibrated rigid structure from motion with a proper perspective projection can be solved within a factorization framework.
On the downside the accuracy of these methods comes with a price.
Typically the iterations are costly since (even when the Schur complement trick is used) 2nd order methods require an inversion of a relatively large hessian matrix, which may hinder application when suitable sparsity patterns are not present.

For low level vision problems such as denoising and inpainting, eg.
\cite{oh-etal-pami-2016,hu-etal-pami-2013,gu-2016}, the main interest is to recover the elements of $X$ and not the factorization. In this context more general regularization terms that also consider the size of the singular values are often used. Since the singular values are non-differentiable functions of the elements in $X$ first order methods are usually employed. The simplest option is perhaps a splitting methods such as ADMM \cite{boyd-etal-2011} since the proximal operator
\begin{equation}
\argmin_X f(\sigma(X)) + \|X-X_0\|^2,
\end{equation}
can often be computed in closed form \cite{hu-etal-pami-2013,gu-2016,oh-etal-pami-2016,dai-etal-ijcv-2014,kumar-arxiv-2019}. Alternatively, subgradient methods can be used to handle the non-differentiability of the regularization term \cite{canyi-etal-cvpr-2014}.

It is well known that while first order methods have rapid iterations and make large improvements the first couple of iterations they have a tendency to converge slowly when approaching the optimum.
For example, \cite{boyd-etal-2011} recommends to use ADMM when a solution in the vicinity of the optimal point is acceptable, but suggests to switch to a higher order method when high accuracy is desired.
For low level vision problems where success is not dependent on achieving an exact factorization of a particular size, first order methods may therefore be suitable. In contrast, in the context of structure from motion, having roughly estimated elements in $X$ causes the obtained factorization $B$, $C$ to be of a much larger size than necessary yielding poor reconstructions with too many deformation modes.

In this paper we aim to extend the class of methods that can be optimized using bilinear parameterization allowing accurate estimation of a low rank factorization from a general class of regularization terms. While our theory is applicable for many objectives we focus on weighted nuclear norm penalties since these have been successfully used in structure form motion applications. We show that these can be optimized with 2nd order methods which significantly increases the accuracy of the reconstruction. We further show that with these improvements the model of Hong \textit{et al}.  \cite{hong-zach-cvpr-2018} can be extended to handle non-rigid reconstruction with a proper perspective model, as opposed to the orthographic projection model adopted by other factorization based approaches, e.g. \cite{kumar-arxiv-2019,dai-etal-ijcv-2014,garg-etal-cvpr-2013,yan-pollefeys-pami-2008}.

\subsection{Related Work and Contributions}

Minimization directly over $X$ has been made popular since the problem is convex when $f$ is convex and absolutely symmetric, that is, $f(|x|) = f(x)$ and $f(\Pi x) = f(x)$, where $\Pi$ is any permutation \cite{lewis1995convex}.
Convex penalties are however of limited interest since they generally prefer solutions with many small non-zero singular values to those with few large ones. A notable exception is the nuclear norm  \cite{fazel-etal-acc-2015,recht-etal-siam-2010,oymak2011simplified,candes-etal-acm-2011,candes2009exact} which penalizes the sum of the singular values. Under the RIP assumption \cite{recht-etal-siam-2010} exact or approximate low rank matrix recovery can then be guaranteed \cite{recht-etal-siam-2010,candes2009exact}.
On the other hand, since the nuclear norm penalizes large singular values, it suffers from a shrinking bias \cite{cabral-etal-iccv-2013,canyi-etal-cvpr-2014,larsson-olsson-ijcv-2016}.

An alternative approach that unifies bilinear parameterization with regularization approaches is based on the observation \cite{recht-etal-siam-2010} that the nuclear norm $\|X\|_*$ of a matrix $X$ can be expressed as
$
\|X\|_* = \min_{BC^T = X} \frac{\|B\|_F^2+\|C\|_F^2}{2}.
$
Thus when $f({\bm \sigma}(X)) = \mu \sum_i\sigma_i(X)$, where $\mu$ is a scalar controlling the strength of the regularization, optimization of \eqref{eq:general} can be formulated as
{\small
\begin{equation}
\min_{B,C} \mu \frac{\|B\|_F^2+\|C\|_F^2}{2}+ \|{\mathcal{A}} BC^T - b\|^2.
\label{eq:nuclearbilin}
\end{equation}}\normalsize
Optimizing directly over the factors has the advantages that the number of variables is much smaller and the objective function is two times differentiable so second order methods can be employed.
While \eqref{eq:nuclearbilin} is non-convex because of the bilinear terms, the convexity of the nuclear norm can still be used to show that any local minimizer $B$,$C$ with ${\text{ rank}}(B C^T) < k$, where $k$ is the number of columns in $B$ and $C$, is globally optimal \cite{bach-arxiv-2013,haeffele-vidal-arxiv-2017}.
The formulation \eqref{eq:nuclearbilin} was for vision problems in \cite{cabral-etal-iccv-2013}. In practice it was observed that the shrinking bias of the nuclear norm makes it too weak to enforce a low rank when the data is noisy. Therefore, a ``continuation'' approach where the size of the factorization is gradually reduced was proposed. While this yields solutions with lower rank, the optimality guarantees no longer apply.
Bach \textit{et al}.  \cite{bach-arxiv-2013} showed that
{\small
\begin{equation}
\|X\|_{s,t}:=\min_{X=BC^T} \sum_{i=1}^k\frac{\|B_i\|_s^2 + \|C_i\|_t^2}{2},
\label{eq:decomposition-norm}
\end{equation}}\normalsize
where $B_i$,$C_i$ are the $i$th columns of $B$ and $C$ respectively,
is convex for any choice of vector norms $\|\cdot\|_s$ and \mbox{$\|\cdot\|_t$}.
In \cite{haeffele-vidal-arxiv-2017} it was shown that a more general class of 2-homogeneous factor penalties result in a convex regularization similar to \eqref{eq:decomposition-norm}. The property that a local minimizer $B$, $C$ with ${\text{ rank}}(B C^T) < k$, is global is also extended to this case. Still, because of convexity, it is clear that these formulations will suffer from a similar shrinking bias as the nuclear norm.

One way of reducing shrinking bias is to use penalties that are constant for large singular values.
Shang \textit{et al}.  \cite{shang-etal-2018} showed that penalization with the Schatten semi-norms $\|X\|_q = \sqrt[q]{\sum_{i=1}^N \sigma_i(X)^q}$, for $q=1/2$ and $2/3$, can be achieved using a convex penalty on the factors $B$ and $C$. A generalization to general values of $q$ is given in \cite{xu-etal-AAAI-2017}.
An algorithm that address a general class of penalties for symmetric matrices is presented in \cite{krechetov-etal-2019}.
In \cite{valtonen-ornhag-etal-2018-arxiv} it was shown that if $f$ is separable with the same penalty for each singular value, that is, $f({\bm \sigma}(X)) = \sum_i g(\sigma_i(X))$, where $g$ is differentiable, concave and non-decreasing then \eqref{eq:general} can be optimized using second order methods such as Levenberg-Marquart or VarPro. This is achieved by re-parameterizing the matrix $X$ using a bilinear factorization $X = B C^T$ and optimizing
\begin{equation}
\min_{B,C} f({\bm \gamma}(B,C))+\|{\mathcal{A}}(BC^T) + b\|^2.
\label{eq:generalbilinear}
\end{equation}
Here ${\bm \gamma}(B,C) = (\gamma_1(B,C),\gamma_2(B,C),...)$ and $\gamma_i(B,C) = \frac{\|B_i\|^2+\|C_i\|^2}{2}$. In contrast to the singular value $\sigma_i(X)$ the function
$\gamma_i(B,C)$ is smooth which allows optimization with second order methods.
It is shown in \cite{valtonen-ornhag-etal-2018-arxiv} that if $X^*$ is optimal in \eqref{eq:general}
then the factorization $B=L\sqrt{\Sigma}, C = R\sqrt{\Sigma}$, where $X^* = L \Sigma R^T$ is the SVD of $X^*$, is optimal in \eqref{eq:generalbilinear}. (Here we assume that $L$ is $m \times r$, $\Sigma$ is $r \times r$ and $R$ is $n\times r$, with ${\text{ rank}}(X)=r$.) Note also that this choice gives $\gamma_i(B,C) = \sigma_i(X^*)$.

Having a separable objective with the same penalty for each singular value is however somewhat restrictive.
An alternative way of reducing bias is to re-weight the nuclear norm and use $f({\bm \sigma}(X)) = \sum_i a_i \sigma_i(X)$ \cite{hu-etal-pami-2013,gu-2016,kumar-arxiv-2019}. Assigning low weights to the first (largest) singular values allows accurate matrix recovery. In addition the weights can be used to regularize the size of the non-zero singular values which has been shown to be an additional useful prior in NRSfM \cite{kumar-arxiv-2019}. Note however that the singular values are always ordered in non-increasing order. Therefore, while the function is linear in the singular values it is in fact non-convex and non-differentiable in the elements of $X$ whenever the singular values are not distinct (typically the case in low rank recovery).

In this paper we show that this type of penalties allow optimization with ${\bm \gamma}(B,C)$ instead of ${\bm \sigma}(X)$.
In particular we study the optimization problem
{\small
\begin{eqnarray}
& \min_{B,C} & f({\bm \gamma}(B,C)) \label{eq:regobj}\\
& \text{s.t.} & BC^T = X, \label{eq:regconst}
\end{eqnarray}}
and its constraint set for a fixed $X$. We characterize the extreme-points of the feasible set using permutation matrices and give conditions on $f$ that ensure that the optimal solution is of the form ${\bm \gamma}(B,C) = \Pi {\bm \sigma}(X)$, where $\Pi$ is a permutation.
For the weighted nuclear norm $f({\bm \sigma}(X)) = a^T {\bm \sigma}(X)$ we show that if the elements of $a$ are non-decreasing the optimal solution has ${\bm \gamma}(B^*,C^*) = {\bm \sigma}(X)$. A simple consequence of this result is that
\begin{equation}
 \min_{B,C} a^T {\bm \gamma}(B,C)) + \|{\mathcal{A}} (B C^T) - b\|^2
 \label{eq:wnnbilinear}
\end{equation}
is equivalent to
\begin{equation}
\min_X a^T {\bm \sigma}(X) + \|{\mathcal{A}} X - b\|^2.
\end{equation}
While the latter is non-differentiable the former is smooth and can be minimized efficiently with second order methods.

Our experimental evaluation confirms that this approach outperforms current first order methods in terms of accuracy as can be expected. On the other hand first order methods make large improvments the first coupler of iterations and therefore we combine the two approaches. We start out with a simple ADMM implementation and switch to our second order approach when only minor progress is being made. Note however that since the original formulation is non-convex local minima can exist. In addition bilinear parameterization introduces additional stationary points that are not present in the original $X$ parameterization. One such example is $(B,C)=(0,0)$, where all gradients vanish. Still our experiments show that the combination of these methods often converge to a good solution from random initialization.

\section{Bilinear Parameterization Penalties}
In this section we will derive a dependence between the singular values $\sigma_i(X)$ and the $\gamma_i(B,C)$, when $BC^T = X$.
For ease of notation we will suppress the dependence on $X$ and $(B,C)$ since this will be clear from the context.
Let $X$ have the SVD $X = R\Sigma L^T$, $B=R\sqrt{\Sigma}$ and $C = L\sqrt{\Sigma}$.
We will study other potential factorizations $X = \hat{B}\hat{C}^T$ using
$\hat{B} = BV$, $\hat{C} = CH$ and $VH^T = I_{r \times r}$.
In this section we will further assume that $V$ is a square $r \times r$ matrix and therefore $H^T$ is its inverse.
(We will generalize the results to the rectangular case in Section~\ref{sec:generalmat}).

We begin by noting that $\gamma_j = \frac{\|\hat{B}_j\|^2+\|\hat{C}_j\|^2}{2} = \frac{\|BV_j\|^2+\|CH_j\|^2}{2}$, where $V_j$ and $H_j$ are columns $j$ of $V$ and $H$ respectively.
We have $\|BV_j\|^2 = V_j^T B^T B V_j = V_j^T \Sigma V_j = \|\sqrt{\Sigma} V_j\|^2,$
and similarly $\|CH_j\|^2=\|\sqrt{\Sigma} H_j\|^2$ and therefore $\gamma_j = \frac{\|\sqrt{\Sigma}V_j\|^2+\|\sqrt{\Sigma}H_j\|^2}{2}$.
This gives
{\small
\begin{equation}
\gamma_j = \left(\frac{\sigma_1(v_{1j}^2+h_{1j}^2)+\sigma_2(v_{2j}^2+h_{2j}^2)+...+\sigma_r(v_{rj}^2+h_{rj}^2)}{2}\right),
\end{equation}}
or in matrix form
{\small
\begin{equation}
\left(
\begin{matrix}
\gamma_1 \\
\gamma_2 \\
\vdots \\
\gamma_r
\end{matrix}
\right) =
\frac{1}{2}\underbrace{\left(
\begin{matrix}
v_{11}^2 & v_{21}^2 & \hdots & v_{r1}^2 \\
v_{12}^2 & v_{22}^2 & \hdots & v_{r2}^2 \\
\vdots & \vdots & \ddots & \vdots \\
v_{1r}^2 & v_{2r}^2 & \hdots & v_{rr}^2 \\
\end{matrix}
\right)}_{=V^T \odot V^T}
\left(
\begin{matrix}
\sigma_1 \\
\sigma_2 \\
\vdots \\
\sigma_r
\end{matrix}
\right)
+
\frac{1}{2}\underbrace{\left(
\begin{matrix}
h_{11}^2 & h_{21}^2 & \hdots & h_{r1}^2 \\
h_{12}^2 & h_{22}^2 & \hdots & h_{r2}^2 \\
\vdots & \vdots & \ddots & \vdots \\
h_{1r}^2 & h_{2r}^2 & \hdots & h_{rr}^2 \\
\end{matrix}
\right)}_{=H^T\odot H^T}
\left(
\begin{matrix}
\sigma_1 \\
\sigma_2 \\
\vdots \\
\sigma_r
\end{matrix}
\right).
\label{eq:matrixform}
\end{equation}}
Minimizing \eqref{eq:regobj} over different factorizations is therefore equivalent to solving
\begin{eqnarray}
&\min_{{\bm \gamma}, M \in \mathcal{S}} & f({\bm \gamma}), \label{eq:Sobj} \\
& \text{s.t.} & {\bm \gamma} = M {\bm \sigma}. \label{eq:Sconst}
\end{eqnarray}
where
{\small
\begin{equation}
\mathcal{S} = \{\frac{1}{2}(V^T\odot V^T + H^T \odot H^T); \ VH^T = I\}.
\end{equation}}
\iffalse
{\small
\begin{eqnarray}
&\min_{{\bm \gamma},V,H}& f({\bm \gamma}), \label{eq:optprobl} \\
& & {\bm \gamma} = \frac{1}{2}(V^T \odot V^T + H^T \odot H^T) {\bm \sigma} \\
& & VH^T = H^T V = I \label{eq:optconst}.
\end{eqnarray}}
\fi
It is clear that $V = H = \Pi^T$, where $\Pi$ is any permutation, is feasible in the above problem since permutations are orthogonal. In addition they contain only zeros and ones and therefore it is easy to see that this choice gives ${\bm \gamma} = \frac{1}{2}(\Pi \odot \Pi + \Pi \odot \Pi) {\bm \sigma} = \Pi {\bm \sigma}$.
In the next section we will show that these are extreme points of the feasible set, in the sense that they can not be written as convex combinations of other points in the set.
Extreme points are important for optimization since the global minimum is guaranteed to be attained (if it exists) in such a point if the objective function has concavity properties.
This is for example true if $f$ is quasi-concave, that is, the super-level sets $S_{\alpha}=\{x\in \mathbb{R}^r_{\geq 0}; f(x)\geq \alpha\}$ are convex. To see this let $x = \lambda x_1+(1-\lambda)x_2$, and consider the super-level set $S_{\alpha}$ where $\alpha = \min(f(x_1),f(x_2))$. Since both $x_1 \in S_{\alpha}$ and $x_2 \in S_{\alpha}$ it is clear by convexity that so is $x$ and therefore $f(x) \geq \min(f(x_1),f(x_2))$.

\subsection{Extreme Points and Optimality}\label{sec:extremepts}
We now consider the optimization problem \eqref{eq:Sobj}-\eqref{eq:Sconst}
and a convex relaxation of the constraint set. For this purpose we let $\mathcal{D}$ be the set of doubly stochastic matrices
{\small
\begin{equation}
\mathcal{D} = \{M\in \mathbb{R}^{r \times r}; \ m_{ij} \geq 0, \ \sum_i m_{ij} = 1, \ \sum_j m_{ij} = 1\}.
\end{equation}}
Note that if $V$ is orthogonal, and therefore $H=V$, then the row sum
$\sum_{j = 1}^r\frac{v_{ij}^2 + h_{ij}^2}{2}$, and the column sum  $\sum_{i = 1}^r\frac{v_{ij}^2 + h_{ij}^2}{2}$ are both one. Hence such a matrix is in $\mathcal{D}$.
To handle non-orthogonal matrices we define the set of superstochastic matrices $\mathcal{S}_W$ as all matrices $M=D+N$, where $D\in \mathcal{D}$ and $N$ is a matrix with non-negative elements.

It can be shown that (see Theorem~6 in~\cite{bhatia-jain-2015}) that $\mathcal{S} \subset \mathcal{S}_W$. In addition it is easy to see that $\mathcal{S}_W$ is convex since it consists of affine constraints.
Therefore the problem
\begin{eqnarray}
&\min_{{\bm \gamma}, M \in \mathcal{S}_W} & f({\bm \gamma}), \label{eq:Cobj}\\
& \text{s.t.} & {\bm \gamma} = M {\bm \sigma}. \label{eq:Cconst}
\end{eqnarray}
is a relaxation of \eqref{eq:Sobj}-\eqref{eq:Sconst}.
We will next show that the two problems have the same minimum if a minimizer to \eqref{eq:Cobj}-\eqref{eq:Cconst} exists when the objective function $f$ is quasi-concave (on $\mathbb{R}^r_{\geq 0}$).
As we have seen previously, the minimum (over $\mathcal{S}_W$) is then attained in an extreme point of $\mathcal{S}_W$.
We therefore need the following characterization.
\begin{lemma}
	The extreme points of $\mathcal{S}_W$ are $r \times r$ permutation matrices.
\end{lemma}
\begin{proof}
First we note that any extreme point of $\mathcal{S}_W$ has to be in $\mathcal{D}$ since if $M=D+N$ with $N \neq 0$ then $M=\frac{1}{2}D+\frac{1}{2}(D+2N)$, which is a convex combination of two points in $\mathcal{S}_W$.
By Birkhoff's Theorem \cite{birkhoff-1946} any matrix in $\mathcal{D}$ can be written as a convex combination of permutation matrices.
\end{proof}

Since permutation matrices are orthogonal with $0/1$ elements it is clear they can be written
$\Pi = \frac{1}{2}(\Pi \odot \Pi + \Pi \odot \Pi), $
with $\Pi \Pi^T = I$. Therefore the extreme points of $\mathcal{S}_W$ are also in $\mathcal{S}$.
Hence if the minimum of \eqref{eq:Cobj}-\eqref{eq:Cconst} is attained, there is an optimal extreme point of $\mathcal{S}_W$ which also solves \eqref{eq:Sobj}-\eqref{eq:Sconst}, and therefore the solution is given by a permutation $V=H=\Pi$.

We conclude this section by giving sufficient conditions for the minimum of \eqref{eq:Cobj}-\eqref{eq:Cconst} to exist, namely that $f$ is lower semi-continuous and  non-decreasing in all of its variables, that is,
if $\tilde{\gamma}_i \geq \gamma_i$ for all $i$ then $f(\tilde{{\bm \gamma}}) \geq f({\bm \gamma})$.
Since the singular values are all positive it is clear that the elements of $(D+N){\bm \sigma}$ are larger than those of $D{\bm \sigma}$. Hence when $f$ is non-decreasing it is enough to consider minimization over $\mathcal{D}$. We then have a lower semi-continuous objective function on a compact set for which the minimum is known to be attained.

We can now summarize the results of this section in the following theorem:
\begin{theorem}
	Let $f$ be quasi-concave (and lower semi-continuous) on $\mathbb{R}^r_{\geq 0}$ fulfilling $f(\tilde{{\bm \gamma}}) \geq f({\bm \gamma})$ when
	$\tilde{\gamma}_i \geq \gamma_i$ for all $i$. Then there is an optimal ${\bm \gamma}^*$ of \eqref{eq:Sobj}-\eqref{eq:Sconst} that is of the form  ${\bm \gamma}^* = \Pi {\bm \sigma}$ where $\Pi$ is a permutation.
\end{theorem}

\section{Non-square Matrices}\label{sec:generalmat}
In the previous section we made the assumption that $V$ and $H$ where square matrices, which corresponds to  searching over $\hat{B}$ and $\hat{C}$ consisting of $r$ columns when ${\text{ rank}}(X) = r$. In addition since $V$ and $H$ are invertible this means that $\hat{B}$ and $\hat{C}$ have linearly independent columns.
In this section we generalize the result from Section~\ref{sec:extremepts} to rectangular matrices $V$ and $H$. Therefore we suppose that $V$ and $H$ are non-square of size $r \times p$, $p > r$, with $VH^T = I_{r \times r}$,
and consider the slightly modified problem
{\small
\begin{eqnarray}
&\min_{{\bm \gamma},V,H}& f({\bm \gamma}), \label{eq:optproblrec} \\
& & {\bm \gamma} = \frac{1}{2}(V^T \odot V^T + H^T \odot H^T) {\bm \sigma} \\
& & VH^T = I_{r \times r} \label{eq:optconstrec}
\end{eqnarray}}
Note that $VH^T$ do not commute and we therefore only assume that $V$ is a left inverse of $H^T$.
In what follows we show that by adding zeros to the vector $\sigma$ we can extend $V$, $H$ into square matrices without changing the objective function.

Note that we may assume that $V$ has full row rank since otherwise $X \neq BVH^TC^T$.
Let $V^\dagger$ be the Moore-Penrose pseudo inverse and $O_{V_\perp}$ a $(p-r) \times p$ matrix containing a basis for the space orthogonal to the row space of $V$ (and the column space of $V^\dagger$).
Since $VH^T = I_{r \times r}$ the matrix $H^T$ is of the form $H^T = V^\dagger +  O^T_{V_\perp}K_1$, where $K_1$ is a $(p-r)\times r$ coefficient matrix.
We now want to find matrices $\tilde{V}$ and $\tilde{H}$ such that
{\small
\begin{equation}
\begin{bmatrix}
V \\
\tilde{V}
\end{bmatrix}
\begin{bmatrix}
V^\dagger +  O^T_{V_\perp}K_1 & \tilde{H}^T
\end{bmatrix}
=
\begin{bmatrix}
I_{r\times r} & 0 \\
0 & I_{(p-r)\times(p-r)}
\end{bmatrix}.
\end{equation}}
To do this we first select $\tilde{H}^T = O_{V_{\perp}}^T$ since $V O_{V_{\perp}}^T = 0$.
Then we let $\tilde{V} = O_{V_\perp} + K_2 V$, where $K_2$ is a size $(p-r)\times r$ coefficient matrix, since this gives $\tilde{V} \tilde{H}^T = I_{(p-r)\times(p-r)}$.
To determine $K_2$ we consider $\tilde{V}(V^\dagger +  O^T_{V_\perp}K_1) = K_2 I_{r \times r} + I_{(p-r)\times(p-r)}K_1 = K_2+K_1$. Selecting $K_2 = -K_1$ thus gives square matrices such that
{\small
\begin{equation}
\begin{bmatrix}
V \\
\tilde{V}
\end{bmatrix}
\begin{bmatrix}
H^T \tilde{H}^T
\end{bmatrix}
= I.
\end{equation}}
Further letting $\tilde{\Sigma} = \begin{bmatrix}
\Sigma & 0 \\
0 & 0
\end{bmatrix}$ shows that $\|B V_i\| = \|\sqrt{\tilde{\Sigma}} \begin{bmatrix}
V_i \\
\tilde{V}_i
\end{bmatrix}\|$ and $\|C H_i\| = \|\sqrt{\tilde{\Sigma}} \begin{bmatrix}
H_i \\
\tilde{H}_i
\end{bmatrix}\|$ and the results of the previous section give that the minimizer of $f(\gamma_1,\gamma_2,...,\gamma_p)$ is a permutation of the elements in the vector $(\sigma_1,\sigma_2,...,\sigma_r,0,...,0)$.

We therefore have the following result:
\begin{corollary}
Let $f$ be quasi-concave (and lower semi-continuous) on $\mathbb{R}^p_{\geq 0}$ fulfilling $f(\tilde{{\bm \gamma}}) \geq f({\bm \gamma})$ when
$\tilde{\gamma}_i \geq \gamma_i$ for all $i$. Then an optimizer ${\bm \gamma}^*$ of \eqref{eq:optproblrec}-\eqref{eq:optconstrec} is of the form  ${\bm \gamma}^* = \Pi_{p \times r} {\bm \sigma}$ where $\Pi_{p \times r}$ contains the first $r$ columns of a $p \times p$ permutation matrix.
\end{corollary}

\section{Linear Objectives - Weighted Nuclear Norms} \label{sec:wnn}

We now consider weighted nuclear norm regularization  $f({\bm \gamma}) = a^T {\bm \gamma}$.
To ensure that the problem is well posed we assume that the elements of $a$ are non-negative. It is then clear that $f(\tilde{{\bm \gamma}}) \geq f({\bm \gamma})$ when $\tilde{\gamma}_i \geq \gamma_i$.
Since linearity implies concavity the results of Sections~\ref{sec:extremepts} and \ref{sec:generalmat} now show that the minimum of $f(M{\bm \sigma})$, over $M\in\mathcal{S}$ is attained in $M = \Pi$ for some permutation matrix.
To ensure that the bilinear formulation is equivalent to the original one we need to show that the optimum occurs when $\Pi = I$.
Suppose that the elements in $a$ are sorted in ascending order $a_1 \leq a_2 \leq ... \leq a_p$.
It is easy to see that for $\Pi$ to give the smallest objective value it should sort the elements of $\gamma$ so that $\gamma_1 \geq \gamma_2 \geq ... \geq \gamma_p$, which means that $\Pi = I$ and ${\bm \gamma}={\bm \sigma}$. We therefore conclude that minimizing \eqref{eq:generalbilinear} with a linear objective corresponds to regularization with a weighted nuclear norm with non-decreasing weights.

\section{Experiments}
In this section we start by describing implementation details of our method and then apply it to the problems of low matrix recovery and non-rigid structure recovery.
Solving the weighted nuclear norm regularized problem \eqref{eq:wnnbilinear} now amounts to minimizing
{\small
\begin{equation}
\sum_{i=1}^p a_i \frac{\|B_i\|^2+\|C_i\|^2}{2} + \|{\mathcal{A}}(BC^T) - b\|^2.
\label{eq:fixweightedproblem}
\end{equation}}
Note that the terms in the objective \eqref{eq:fixweightedproblem} can be combined into a single norm term by vertically concatenating the vectors $B_i$ and $C_i$, weighted by $\sqrt{a_i/2}$, with ${\mathcal{A}}(BC^T) - b$. We define the resulting vector as $r_a := {\mathcal{A}}_a(BC^T) - b_a$, giving the objective  $\|r_a(BC^T)\|^2$, where the subscript reflects the dependence on the weights $a$.
Since the objective is smooth, standard methods such as Levenberg-Marquardt can be applied and Algorithm~\ref{alg:bilinear_param_wnn} shows an overview of the method used. Additional information about the algorithm is provided in the supplementary material.

The remainder of this section is organized as follows.
The particular form of the data fitting term in \eqref{eq:fixweightedproblem} when applied to structure from motion is described in Section~\ref{sec:pose_error}.
In Section~\ref{sec:exp_matrix_recovery} we compare the convergence of first and second-order methods, and motivated by the ADMM fast iterations but low accuracy, as opposed to the bilinear parameterization's high accuracy but slower iterations, we combine the two methods by initializing the bilinear parameterization with the ADMM's solution \cite{boyd-etal-2011,kumar-arxiv-2019} for a non-rigid structure structure recovery problem. Our work focus on the increased accuracy of our method compared to first-order methods, so the comparison of our results with  works such as \cite{xiao_2004,yan_2008,akhter_2008,paladini_2009} (without the desired regularization term) are not covered.

\begin{algorithm}[t]
\algsetup{linenosize=\small}
  \scriptsize
\SetAlgoLined
\KwResult{Optimal B,C to \eqref{eq:fixweightedproblem}}
 $B = U\Sigma^{\frac{1}{2}}$, $C = V\Sigma^{\frac{1}{2}}$, where $X = U \Sigma V^T$ is the ADMM solution to \eqref{eq:fixweightedproblem} \;
 Choose initial $\alpha > 1$ and $\lambda$, and define $z = [\mathrm{vec}(B);\text{ }\mathrm{vec}(C^T)]$\;
 Compute \textit{error} $=\|r_a(BC^T)\|^2$\;
 \While{not converged}{
 Compute $r = r_a(BC^T)$, and the jacobian $J$ of \eqref{eq:fixweightedproblem} in terms of $z$\;
  Update $\tilde{z} = z - (J^TJ + \lambda I)^{-1} J^Tr$, where $\tilde{z} = [\mathrm{vec}(\tilde{B});\text{ }\mathrm{vec}(\tilde{C}^T)]$\;
  \eIf{error $>\|r_a(\tilde{B}\tilde{C}^T)\|^2$ }{
  Updata $z \leftarrow \tilde{z}$,
   $\lambda \leftarrow \alpha^{-1} \lambda$, and
   \textit{error} $\leftarrow \|{\mathcal{A}}_a(\tilde{B}\tilde{C}^T) - b_a\|^2$\;
   }{
   $\lambda \leftarrow \alpha \lambda$\;
  }
 }
 \caption{Bilinear parameterization of weighted nuclear norm}
 \label{alg:bilinear_param_wnn}
\end{algorithm}

\subsection{Pseudo Object Space Error (pOSE)  and Non-Rigid Structure from Motion}
\label{sec:pose_error}

To compare the performance of first and second-order methods, we choose as objective function the Pseudo Object Space Error (pOSE) \cite{hong-zach-cvpr-2018}, which consists of a combination of the object space error $\ell_{\mathrm{OSE}} := \sum_{(i,j)\in \Omega}\| \mathrm{P}_{i,1:2}\tilde{\bm{\mathrm{x}}}_j - (\bm{\mathrm{p}}_{i,3}^T\tilde{\bm{\mathrm{x}}}_j)\bm{\mathrm{m}}_{i,j} \|^2_2$
and the affine projection error
    $\ell_{\mathrm{Affine}} := \sum_{(i,j)\in \Omega}\| \mathrm{P}_{i,1:2}\tilde{\bm{\mathrm{x}}}_j - \bm{\mathrm{m}}_{i,j} \|^2_2$,
where $\mathrm{P}_{i,1:2}$ and $\bm{\mathrm{p}}_{i,3}$ are, respectively, the first two and the third rows of the camera matrix $\mathrm{P}_{i}$, with $i = 1,\hdots,F$; $\tilde{\bm{\mathrm{x}}}_j$ is a 3D point in homogeneous coordinates, with $j = 1,\hdots,P$; $\bm{\mathrm{m}}_{i,j}$ is the 2D observation of the $j$:th point on the $i$:th camera; and $\Omega$ represents the set of observable data. The Pseudo Object Space Error is then given by
\begin{equation}
    \ell_{\mathrm{pOSE}} := (1-\eta)\ell_{\mathrm{OSE}} + \eta \ell_{\mathrm{Affine}}
\end{equation}
where $\eta \in [0,1]$ balances the weight between the two errors. One of the main properties of pOSE is its wide basin of convergence
while keeping a bilinear problem strucuture.
The $\ell_{\mathrm{pOSE}}$, originally designed for rigid structure from motion, can be extended for the non-rigid case by replacing $\mathrm{P}_i\tilde{\bm{\mathrm{x}}}_j$ by a linear combination of $K$ shape basis, i.e.,  $\mathrm{\Pi}_i\hat{\mathrm{S}}_j$, where $\mathrm{\Pi}_i \in \mathbb{R}^{3 \times (3K+1)}$ and $\hat{\mathrm{S}}_j \in \mathbb{R}^{3K+1}$ are structured as
\begin{equation}
    \mathrm{\Pi}_i = \begin{bmatrix}
    c_{i,1}R_i \quad \hdots \quad c_{i,K}R_i \quad t_i
    \end{bmatrix} \quad \text{and} \quad
    \hat{\mathrm{S}}_j = \begin{bmatrix}
    S_{1,j}^T \quad \hdots \quad S_{K,j}^T \quad 1
    \end{bmatrix}^T.
    \label{eq:nonrigid_fact}
\end{equation}
We denote by $\mathrm{\Pi}$ and $\hat{\mathrm{S}}$ the vertical and horizontal concatenations of $\mathrm{\Pi}_i$ and $\hat{\mathrm{S}}_j$, respectively.
Note that for $K = 1$, we have $\mathrm{\Pi}_i\hat{\mathrm{S}}_j = \mathrm{P}_i\tilde{\bm{\mathrm{x}}}_j$, which corresponds to the rigid case, and consequentely $\mathrm{rank}(\mathrm{\Pi}\hat{\mathrm{S}}) = 4$. For $K>1$, by construction we have $\mathrm{rank}(\mathrm{\Pi}\hat{\mathrm{S}}) \leq 3K+1$.

\subsection{Low-Rank Matrix Recovery with pOSE errors}
\label{sec:exp_matrix_recovery}
In this section we compare the performance of first and second-order methods in terms of convergence and accuracy, starting from the same initial guess, for low-rank matrix recovery with pOSE. In this problem, we define $X = \mathrm{\Pi}\hat{\mathrm{S}}$ and aim at minimizing
\begin{equation}
\label{prob:low_rank_matrix_recovery}
\min_X a^T {\bm \sigma}(X) + \ell_{\mathrm{pOSE}}(X),
\end{equation}We apply our method and solve the optimization problem defined in
\eqref{eq:fixweightedproblem} by using the bilinear factorization $X = BC^T$, with $B \in \mathbb{R}^{3F \times r}$, and $C \in \mathbb{R}^{P \times r}$, with $r \geq 3K+1$. We test the performance of our method in 4 datasets: Door \cite{olsson-engqvist-scia-2011}, Back \cite{Russell2011}, Heart \cite{stoyanov2005}, Paper \cite{Varol2009}. The first one consists of image measurements of a rigid structure with missing data, while the remaining three datasets track points in deformable structures.

For the Door dataset, we apply two different selections of weights on the singular values of $X$, corresponding to the nuclear norm, i.e., $a_i = \mu_{NN}$, and truncated nuclear norm, i.e., $a_i = 0$, $i = 1,\hdots,4$ and $a_i = \mu_{TNN}$, $i > 4$. We select $\mu_{NN} = 1.5\times 10^{-3}$, and $\mu_{TNN} = 1$.
For the Back, Heart and Paper datasets, we apply the nuclear norm and a weighted nuclear norm, in which the first four singular values of $X$ are not penalized and the remaining ones are increasingly penalized, i.e., $a_i = 0$, $i = 1,\hdots,4$ and $a_i = (i-4)\mu_{WNN}$, $i > 4$.
We select $\mu_{NN} = 7.5\times 10^{-4}$, $\mu_{WNN} = 2.25\times 10^{-3}$. The values of the weights $a_i$ are chosen such that there is a $3K + 1$ rank solution to \eqref{prob:low_rank_matrix_recovery}, with $K = 1$ and $K=2$ for the rigid and non-rigid datasets, respectively.

We compare the bilinear parameterization with three first-order methods commonly used for low-rank matrix recovery: Alternating Direction Method of Multipliers (ADMM) \cite{boyd-etal-2011}, Iteratively Reweighted Nuclear Norm (IRNN) \cite{canyi2015}, and Accelerated Gradient Descend (AGD) \cite{li2019}. We also test the methods for two different cases of the $\ell_{\mathrm{pOSE}}$ error, with $\eta = 0.05$ and $\eta = 0.95$, which correspond to the near-perspective and near-affine camera models, respectively. To improve numerical stability of the algorithms, as pre-processing step we normalize the image measurements matrix $M$ by its norm. The methods are initialized with the closed-form solution of the regularization-free problem, i.e., $X = {\mathcal{A}}^\dagger(b)$. The comparison of the four algorithms in terms of total log-loss over time is shown in Figure~\ref{fig:exps_matrix_recovery_loss}. The log-loss is used for better visualization purposes. The plots for the IRNN for the nuclear norm are omitted since it demonstrated slow convergence compared to the remaining three methods. A qualitative evaluation of the results on one of the images of the Door dataset for the truncated nuclear norm and near perspective camera model is shown in Figure~\ref{fig:door_eval}.
The qualitative results for the remaining datasets are provided in the supplementary material

In general, we can observe that first-order methods demonstrate faster initial convergence, mostly due to faster iterations. However when near minima, the convergence rate drops significantly and the methods tend to stall. Contrarily, bilinear parameterization compensates its slower iterations by demonstrating higher accuracy and and reaching solutions with lower energy. This is specially visible for the near perspective camera model, which reinforces the advantages of using a second-order method on image data under perspective projection. To compensate for the slower convergence, we propose the initialization of the bilinear parameterization with the solution obtained with ADMM. In this way, the bilinear parameterization starts near the minimum and performs local refinement to further improve accuracy.

\begin{figure}
\centering
\begin{subfigure}{\textwidth}
 \centering
\includegraphics[width=0.47\linewidth]{images/door_exps_02.png}
\hfill
 \includegraphics[width=0.47\linewidth]{images/door_exps_13.png}
\caption{Door Dataset.}
\centering
\end{subfigure}
\begin{subfigure}{\textwidth}
 \centering
\includegraphics[width=0.47\linewidth]{images/heart_exps_02.png}
\hfill
 \includegraphics[width=0.47\linewidth]{images/heart_exps_13.png}
\caption{Heart Dataset.}
\centering
\end{subfigure}
\begin{subfigure}{\textwidth}
 \centering
\includegraphics[width=0.47\linewidth]{images/back_exps_02.png}
\hfill
 \includegraphics[width=0.47\linewidth]{images/back_exps_13.png}
\caption{Back Dataset.}
\centering
\end{subfigure}
\begin{subfigure}{\textwidth}
 \centering
\includegraphics[width=0.47\linewidth]{images/paper_exps_02.png}
\hfill
 \includegraphics[width=0.47\linewidth]{images/paper_exps_13.png}
\caption{Paper Dataset.}
\centering
\end{subfigure}
\caption{Convergence of the four methods for low-rank matrix recovery on the Door, Heart, Back and Paper datasets. }
\label{fig:exps_matrix_recovery_loss}
\end{figure}

\begin{figure}
\centering
\begin{subfigure}{\textwidth}
 \centering
\includegraphics[height=3.5cm]{images/door_orig.png}
\includegraphics[width=0.19\linewidth]{images/door_admm.png}
\includegraphics[width=0.1925\linewidth]{images/door_irnn.png}
\includegraphics[width=0.19\linewidth]{images/door_agd.png}
\includegraphics[width=0.19\linewidth]{images/door_ours.png}
\centering
\end{subfigure}
\caption{Evaluation of the four methods for low-rank matrix recovery on one of the images of the Door dataset. The red circles show the target image measurements and the green circles the estimate image points. }
\label{fig:door_eval}
\end{figure}

\begin{figure}
\centering
\begin{subfigure}{0.15\textwidth}
\end{subfigure}
\begin{subfigure}{0.03\textwidth}
\rotatebox[origin=t]{90}{Articulated}
\end{subfigure}
\begin{subfigure}{0.78\textwidth}
\hspace{.25cm}
\includegraphics[height=2.275cm]{images/articulated.png}\hspace{1.5cm}
\includegraphics[height=2.25cm]{images/articulated_3D.png}
\end{subfigure}

\begin{subfigure}{0.15\textwidth}
\end{subfigure}
\begin{subfigure}{0.03\textwidth}
\rotatebox[origin=t]{90}{Balloon}
\end{subfigure}
\begin{subfigure}{0.78\textwidth}
\hspace{.235cm}
\includegraphics[height=2.275cm]{images/balloon.png}
\hspace{1.5cm}\includegraphics[height=2.25cm]{images/balloon_3D.png}
\end{subfigure}

\begin{subfigure}{0.15\textwidth}
\end{subfigure}
\begin{subfigure}{0.03\textwidth}
\rotatebox[origin=t]{90}{Paper}
\end{subfigure}
\begin{subfigure}{0.78\textwidth}
\hspace{.325cm}\includegraphics[height=2.25cm]{images/paper.png}
\hspace{1.5cm}\includegraphics[height=2.25cm]{images/paper_3D.png}
\end{subfigure}

\begin{subfigure}{0.15\textwidth}
\end{subfigure}
\begin{subfigure}{0.03\textwidth}
\rotatebox[origin=t]{90}{Stretch}
\end{subfigure}
\begin{subfigure}{0.78\textwidth}
\hspace{.32cm}\includegraphics[height=2.25cm]{images/stretch.png}
\hspace{1.5cm}\includegraphics[height=2.25cm]{images/stretch_3D.png}
\end{subfigure}

\begin{subfigure}{0.15\textwidth}
\end{subfigure}
\begin{subfigure}{0.03\textwidth}
\rotatebox[origin=t]{90}{Tearing}
\end{subfigure}
\begin{subfigure}{0.78\textwidth}
\hspace{.325cm}\includegraphics[height=2.25cm]{images/tearing.png}
\hspace{1.5cm}\includegraphics[height=2.25cm]{images/tearing_3D.png}
\end{subfigure}
\caption{(Left) Example of the non-rigid objects in the 5 datasets of the NRSfM Challenge. (Right) Estimation (blue) and ground-truth (red) of the non-rigid 3D structure for the two methods with weighted nuclear norm regularization.}
\label{fig:nrsfm_results}
\end{figure}

\iffalse
\begin{figure}
\centering
\includegraphics[height=3.5cm]{images/hybrid.png}
\caption{
}
\end{figure}
\fi

\subsection{Non-Rigid Structure Recovery}
Consider now that the camera rotations in \eqref{eq:nonrigid_fact} are known (or previously estimated). In this case we have $ \mathrm{\Pi}\hat{\mathrm{S}} = RX+t\mathbbm{1}^T$, with $R = \mathrm{blkdiag}(R_1,\hdots,R_F)$ and $t = [t_1^T,\hdots,t_F^T]^T$, where $X$, the non-rigid structure, and $t$ are the unknowns. It is directly observed that $\mathrm{rank}(\mathrm{\Pi}\hat{\mathrm{S}})\leq \mathrm{rank}(RX) + \mathrm{rank}(t\mathbbm{1}^T)$, with the later being equal to 1 by construction and independent on $K$. As consequence, it follows that $\mathrm{rank}(RX) = \mathrm{rank}(X) \leq 3K$, and the rank regularization can be applied on $X$. A similar problem was studied in \cite{dai-etal-ijcv-2014} but for orthogonal camera models, where the authors propose the rank regularization to be applied on a reshaped version of $X$, a $F \times 3P$ matrix structured as
{\small
\begin{equation}
    X^\# = \begin{bmatrix} X_{1,1} & \hdots & X_{1,P} & Y_{1,1} & \hdots & Y_{1,P} & Z_{1,1} & \hdots & Z_{1,P}  \\
    \vdots & & \vdots & \vdots & & \vdots & \vdots & & \vdots \\
    X_{F,1} & \hdots & X_{F,P} & Y_{F,1} & \hdots & Y_{F,P} & Z_{F,1} & \hdots & Z_{F,P}
    \end{bmatrix} = g^{-1}(X),
\end{equation}}\normalsize
where $X_{i,j}$, $Y_{i,j}$ and $Z_{i,j}$ are respectively the x, y and z-coordinates of the $j$:th 3D point at frame $i$. The function $g$ performs the permutation on the elements of $X^\#$ to obtain $X$. With this reshaping we have that $\mathrm{rank}(X^\#) \leq K$, meaning that we can factorize it as $X^\# = BC^T$ with $B \in \mathbb{R}^{F \times K}$ and $C \in \mathbb{R}^{3P \times K}$. The optimization problem then becomes
{\small
\begin{equation}
\min_{B,C,t} \sum_{i=1}^K a_i \frac{\|B_i\|^2+\|C_i\|^2}{2} + \ell_{\mathrm{pOSE}}(Rg(BC^T)+t\mathbbm{1}^T).
\label{eq:nonrigid_structure_opt}
\end{equation}}Solving this optimization problem requires small adjustments to be done to the proposed Algorithm~\ref{alg:bilinear_param_wnn}, which can be consulted in the supplementary material.
We apply our methods to the 5 datasets (Articulated, Balloon, Paper, Stretch, Tearing) from the NRSfM Challenge \cite{jensen2018}. Each of these datasets include tracks of image points for orthogonal and perspective camera models for six different camera paths (Circle, Flyby, Line, Semi-circle, Tricky, Zigzag), as well as the ground-truth 3D structure for one of the frames. We use the 2D observation for the orthogonal camera model to compute the rotation matrix $R$, as done in \cite{dai-etal-ijcv-2014}, and the ground-truth 3D structure to estimate the intrinsic camera matrix, which is assumed to be fixed during each sequence. The intrinsic camera matrix is used to obtain the calibrated 2D observation of the perspective camera model data. For the nuclear norm (NN), we set $a_i = 1\times 10^{-3}, i = 1,\hdots,K$. For the weighted nuclear norm (WNN), the weights $a$ are selected similarly to \cite{kumar-arxiv-2019}
{\small
\begin{equation}
    a_i = \frac{\xi}{\sigma_i(g^{-1}(X_0)) + \gamma}, \quad i = 1,\hdots,K
\end{equation}}
where $\xi = 5\times 10^{-3}$, $\gamma$ is a small number for numerical stability, and $X_0$ is the closed-form solution of the objective $\min_{X}\ell_{\mathrm{pOSE}}(RX). $

For these datasets we choose $K = 2$ and set the $\eta = 0.05$. As baseline we use the best performing first-order method according to the experiments Section~\ref{sec:exp_matrix_recovery}, ADMM, and apply the method described in Algorithm~\ref{alg:bilinear_param_wnn} for local refinement starting from the ADMM's solution. We also try our method for the orthogonal camera model (by setting $\eta = 1$), and compare it with BMM \cite{dai-etal-ijcv-2014} and R-BMM \cite{kumar-arxiv-2019}, which correspond to ADMM implementations for nuclear norm and weighted nuclear norm, respectively. These methods perform a best rank $K$ approximation to the obtained ADMM solution if $\mathrm{rank}(X^\#) > K$ after convergence. We let the ADMM-based methods run until convergence or stalling is achieved for fair comparison. The average log-losses, before and after refinement, obtained on each dataset over the 6 camera paths are shown in Table~\ref{tab:nrsfm_data_results_loss}. The average reconstruction errors, in millimeters, on each dataset over the 6 paths relatively to the provided ground-truth structure are shown in Table~\ref{tab:nrsfm_data_results}. In Figure~\ref{fig:nrsfm_results} we also show some qualitative results of the obtained 3D reconstruction of each of the objects in the 5 datasets. More qualitative results are provided in the supplementary material.

The results show that our method is able to achieve lower energies for all datasets comparatively with the ADMM baselines. As expected from the experiments in Section~\ref{sec:exp_matrix_recovery}, the difference is more substantial for the perspective model. Furthermore, even though we are not explicitly minimizing the reconstruction error expressed in Table~\ref{tab:nrsfm_data_results}, we are able to consistently obtain the lowest reconstruction error for all datasets, sometimes with great improvements compared to the ADMM (see Balloon and Stretch in Figure~\ref{fig:nrsfm_results}). The same does not apply for the orthogonal data, where achieving lower energies did not lead to lower reconstruction errors.

\small{
\begin{table}[t]
\caption{Average log-loss on each of the perspective datasets over the 6 camera paths.}
\label{tab:nrsfm_data_results_loss}
\begin{tabular}{>{\centering}p{0.13\textwidth}>{\centering}p{0.27\textwidth}>{\centering}p{0.12\textwidth}>{\centering}p{0.12\textwidth}>{\centering}p{0.12\textwidth}>{\centering}p{0.12\textwidth}>{\centering\arraybackslash}p{0.12\textwidth}}
  &\multicolumn{1}{c|}{Method \textbackslash Dataset} & Articulated & Balloon & Paper & Stretch & Tearing \\ \cline{2-7}
\multirow{4}{*}{Orthogonal}
 & \multicolumn{1}{c|}{BMM \cite{dai-etal-ijcv-2014}} & -1.645 & -2.267 & -1.712 & -2.282 & -1.453 \\
   &                     \multicolumn{1}{c|}{Ours-NN} & \textbf{-1.800} & \textbf{-2.352} & \textbf{-2.188} & \textbf{-2.509} & \textbf{-1.634} \\\cdashline{2-7}
  & \multicolumn{1}{c|}{R-BMM \cite{kumar-arxiv-2019}} & -1.648 & -1.979 & -1.855 & -1.997 & -1.522 \\

   & \multicolumn{1}{c|}{Ours-WNN} & \textbf{-1.648} & \textbf{-1.979} & \textbf{-1.855} & \textbf{-1.997} & \textbf{-1.522} \\\cline{2-7}
   \multirow{4}{*}{Perspective}
&\multicolumn{1}{c|}{ADMM-NN} & -2.221 & -2.529 & -2.338 & -2.395 & -1.471 \\
  &\multicolumn{1}{c|}{Ours-NN} & \textbf{-2.415} & \textbf{-2.657} & \textbf{-2.560} & \textbf{-2.622} & \textbf{-2.053} \\\cdashline{2-7}

 & \multicolumn{1}{c|}{ADMM-WNN} & -2.455 & -2.617 & -2.195 & -2.651 & -1.688 \\

  &\multicolumn{1}{c|}{Ours-WNN} & \textbf{-2.486} & \textbf{-2.931} & \textbf{-2.777} & \textbf{-2.857} &  \textbf{-2.103}
\end{tabular}
\end{table}}
\tiny{
\begin{table}[t]
\caption{Average reconstruction errors, in millimeters, on each dataset over the 6 camera paths relatively to the provided ground-truth structure.}
\label{tab:nrsfm_data_results}
\begin{tabular}{>{\centering}p{0.13\textwidth}>{\centering}p{0.27\textwidth}>{\centering}p{0.12\textwidth}>{\centering}p{0.12\textwidth}>{\centering}p{0.12\textwidth}>{\centering}p{0.12\textwidth}>{\centering\arraybackslash}p{0.12\textwidth}}
 & \multicolumn{1}{c|}{Method \textbackslash Dataset} & Articulated & Balloon & Paper & Stretch & Tearing \\ \cline{2-7}
\multirow{4}{*}{Orthogonal}
 & \multicolumn{1}{c|}{BMM \cite{dai-etal-ijcv-2014}} & 18.49 & 10.39 & \textbf{8.94} & 10.02 & \textbf{14.23}\\
   & \multicolumn{1}{c|}{Ours-NN} & 18.31 & 8.53 & 10.94 & 10.67 & 17.03 \\
  & \multicolumn{1}{c|}{R-BMM \cite{kumar-arxiv-2019}} & 16.00 & \textbf{7.84} & 10.69 & \textbf{7.53} &  16.34 \\

   & \multicolumn{1}{c|}{Ours-WNN} & \textbf{15.03} & 8.05 & 10.45 & 9.01 & 16.20 \\\cline{2-7}
\multirow{4}{*}{Perspective}
& \multicolumn{1}{c|}{ADMM-NN} & 16.70 & 8.05 & 7.96 & 6.04 & 9.40 \\

 & \multicolumn{1}{c|}{Ours-NN} & \textbf{16.13} & 6.48 & 6.80 & 6.00 & 9.31 \\

 & \multicolumn{1}{c|}{ADMM-WNN} & 18.33 & 8.95 & 10.14 & 8.06 & 9.28 \\

 & \multicolumn{1}{c|}{Ours-WNN} & 16.53 & \textbf{6.27} & \textbf{5.68} & \textbf{5.93} & \textbf{8.42}
\end{tabular}
\end{table}}
\normalsize

\section{Conclusions}
In this paper we showed that it is possible to optimize a general class of singular value penalties using a bilinear parameterization of the matrix. We showed that with this parameterization weighted nuclear norm penalties turn in to smooth objectives that can be accurately solved with 2nd order methods. Our proposed approach starts by using ADMM which rapidly decreases the objective the first couple of iterations and switches to Levenberg-Marquardt when ADMM iterations make little progress. This results in a much more accurate solution and we showed that we were able to extend the recently proposed pOSE \cite{hong-zach-cvpr-2018} to handle non-rigid reconstruction problems.

While 2nd order methods offer increased accuracy, our approach is expensive since iterations require the inversion of a large matrix. Exploring feasible alternatives such as preconditioning and conjugate gradient approaches is an interesting future direction.

Something that we have not discussed is adding constraints on the factors, which is possible since these are present in the optimization. This is very relevant for structure from motion problems and will likely be an fruitful direction to explore.

\iffalse
\begin{figure}
\centering
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{images/door_detail.png}
\centering
  \label{fig:door_example}
\end{subfigure}
\begin{subfigure}{0.8\textwidth}
 \includegraphics[width=\textwidth]{images/heart_example.png}
\centering
  \label{fig:heart_example}
\end{subfigure}
\begin{subfigure}{0.8\textwidth}
 \includegraphics[width=\textwidth]{images/back_example.png}
\centering
  \label{fig:back_example}
\end{subfigure}
\begin{subfigure}{0.8\textwidth}
 \includegraphics[width=\textwidth]{images/paper_example.png}
\centering
  \label{fig:paper_example}
\end{subfigure}
\caption{plots of....}
\label{fig:fig}
\end{figure}
\fi

\iffalse
\begin{center}
\begin{tabular}{ |c |c| c| c |c| c| c| }
\hline
 Method & PTA & CSF2 & PND & BMM & Kumar & Ours \\
 \hline
 Drink & 0.0287 & 0.0227 & 0.0037 & 0.0266 & 0.0119 & 0.0366 \\
\hline
 Pickup & 0.1939 & 0.1791 & 0.0372 & 0.1731 & 0.0198 & 0.0717 \\
 \hline
 Yoga & 0.1243 & 0.1179 & 0.0140 & 0.1150 & 0.0129 & 0.0414 \\
 \hline
 Stretch & 0.1035 & 0.1136 & 0.0156 & 0.1034 & 0.0144 & 0.0795 \\
 \hline
 Dance & 0.2426 & 0.1877 & 0.1454 & 0.1864 & 0.1060 & ? \\
 \hline
 Walking & 0.3761 & 0.1938 & 0.0465 &  0.1298 & 0.0882 & ? \\
 \hline
 Face & 0.0489 & 0.0319 & 0.0165 & 0.0303 & 0.0179 & ? \\
 \hline
 Shark & 0.2933 & 0.1117 & 0.0135 & 0.2311 & 0.0551 & ? \\
 \hline
\end{tabular}
\end{center}
\fi

\small

\section{More details about Algorithm 1 and its implementation}

In the main paper we propose the minimization of the objective
\begin{equation}
\label{eq:sup_obj}
    \min_X a^T {\bm \sigma}(X) + \|{\mathcal{A}} X - b\|^2
\end{equation}
through a bilinear parameterization $X = BC^T$ and using second-order optimization methods such as   Levenberg-Marquardt. In Section 5 we provide an overview of the algorithm used, and in this section of the supplementary material we provide more details that were omitted from the main text, in particular how to formulate the problems regarding Low-rank Matrix Recovery (Section 5.2) and Non-rigid Structure Recovery (Section 5.3) using the pOSE error introduced in Section 5.1.

We start by showing how the pOSE term in \eqref{eq:sup_obj} can be written as linear mapping of the elements of $X$, resulting in the equivalent objective
\begin{equation}
\label{eq:sup_obj2}
    \min_X a^T {\bm \sigma}(X) + \| A_X\vec{X} - b_X\|^2.
\end{equation}
The terms $\ell_{\mathrm{Affine}}$ and $\ell_{\mathrm{OSE}}$ of the pOSE can be written as
\begin{equation}
\label{eq:l_affine_vec}
    \ell_{\mathrm{Affine}} = \|\Gamma_{1:2}X - M\|_F^2 = \|(I \otimes \Gamma_{1:2})\vec{X} - \mathbf{m}\|^2
\end{equation}
and
\begin{equation}
\label{eq:l_ose_vec}
    \ell_{\mathrm{OSE}} = \|\Gamma_{1:2}X - \Gamma_3X \odot M\|^2_F = \|\left((I \otimes \Gamma_{1:2}) - {\text{ diag}}(\mathbf{m})(I \otimes \Gamma_3)\right)\vec{X} \|^2,
\end{equation}
where the matrices $\Gamma_{1:2},\Gamma_3 \in \mathbbm{R}^{2F \times 3F}$ select the desired rows of $X$ and $M \in \mathbbm{R}^{2F \times P}$ gathers all the 2D observations $\bm{\mathrm{m}}_{i,j}$ with $i =1,\hdots,F$ and $j = 1,\hdots,P$. We define $\bm{\mathrm{m}} = \vec{M}$. The rows $2i-1$ and $2i$ of $\Gamma_{1:2}X$ are equal to the rows $3i-2$ and $3i-1$ of $X$, respectively. The rows $2i-1$ and $2i$ of $\Gamma_3X$ are both equal to the row $3i$ of $X$. To obtain \eqref{eq:l_affine_vec} and \eqref{eq:l_ose_vec} we use $\vec{AXB} = (B^T \otimes A)\vec{X}$, where $\otimes$ denotes the Kronocker product.

This allows us to write $A_X$ and $b_X$ in \eqref{eq:sup_obj2} as
\begin{equation}
    A_X = \begin{bmatrix} \sqrt{\eta}\left(I \otimes \Gamma_{1:2}\right)\\
    \sqrt{1-\eta}\left((I \otimes \Gamma_{1:2}) - {\text{ diag}}(\mathbf{m})(I \otimes \Gamma_3)\right)
    \end{bmatrix},\quad
    b_X = \begin{bmatrix}  \sqrt{\eta}\mathbf{m}\\
    \mathbf{0}
    \end{bmatrix}.
\end{equation}
We use this as starting point to formulate the problems of Low-rank Matrix Recovery and Non-rigid Structure Recovery, which differ on the way $X$ in \eqref{eq:sup_obj2} is parameterized.

\subsection{Low-rank Matrix Recovery with pOSE errors}
As seen in Section 5.2, we parameterize $X = BC^T$, which results in the objective
\begin{equation}
\sum_{i=1}^p a_i \frac{\|B_i\|^2+\|C_i\|^2}{2} + \|A_X\vec{BC^T} - b_X\|^2.
\label{eq:sup_fixweightedproblem}
\end{equation}
The pOSE term in \eqref{eq:sup_fixweightedproblem} is no longer linear in $B$ and $C$, and in order to apply Levenberg-Marquardt method we linearize it in the neighbourhood of $B_0$ and $C_0$ as
\begin{equation}
     A_X\vec{BC^T} - b_X \approx \left(A_X\vec{B_0C_0^T} - b_X \right) + A_B\vec{\delta B} + A_{C^T}\vec{\delta C^T}
\end{equation}
where we define
\begin{equation}
    r_{\mathrm{pOSE}} = A_X\vec{B_0C_0^T} - b_X
\end{equation}
with
\begin{equation}
\label{eq:ab_ac_matrices}
    A_B = A_X(C_0 \otimes I), \quad A_{C^T} = A_X(I \otimes B_0).
\end{equation}
 The terms corresponding to the weighted nuclear norm can also be written in a similar fashion since we have
\begin{equation}
    \sum_i \frac{a_i}{2}\|B_i\|^2 = \|B{\text{ diag}}(\sqrt{a/2})\|_F^2 = \|({\text{ diag}}(\sqrt{a/2}) \otimes I )\vec{B}\|^2,
\end{equation}
\begin{equation}
    \sum_i \frac{a_i}{2}\|C_i\|^2 = \|{\text{ diag}}(\sqrt{a/2})C^T\|_F^2 = \|(I \otimes {\text{ diag}}(\sqrt{a/2}))\vec{C^T}\|^2.
\end{equation}
Again, by considering the deviations from the current estimations, $B = B_0 + \delta B$ and $C = C_0 + \delta C$, we end up with
\begin{equation}
\label{eq:sup_regB}
    A_{\mathrm{regB}} = {\text{ diag}}(\sqrt{a/2}) \otimes I, \quad r_{\mathrm{regB}} = ({\text{ diag}}(\sqrt{a/2}) \otimes I )\vec{B_0},
\end{equation}
\begin{equation}
\label{eq:sup_regC}
    A_{\mathrm{regC}} = I \otimes {\text{ diag}}(\sqrt{a/2}), \quad r_{\mathrm{regC}} = (I \otimes {\text{ diag}}(\sqrt{a/2}))\vec{C_0^T}.
\end{equation}
As so, we can compute the residuals and jacobian in Algorithm 1 for the Low-rank Matrix Recovery problem as
\begin{equation}
    J = \begin{bmatrix} A_B & A_{C^T}\\
    A_{\mathrm{regB}} & \mathbf{0}\\
    \mathbf{0} & A_{\mathrm{regC}}
    \end{bmatrix}, \quad r_a = {\mathcal{A}}_a(B_0C_0^T) + b_a = \begin{bmatrix}
    r_{\mathrm{pOSE}}\\r_{\mathrm{regB}}\\r_{\mathrm{regC}}
    \end{bmatrix}.
\end{equation}

\subsection{Non-Rigid Structure Recovery}
When considering the Non-rigid Structure Recovery problem, we use the parameterization $X = Rg(BC^T)+t\mathbbm{1}^T$. This also results in a non-linear pOSE term in terms of $B$ and $C$, and its linearization around $B_0$, $C_0$ and $t_0$ are obtained as
\begin{multline}
     A_X\vec{Rg(BC^T)+t\mathbbm{1}^T} - b_X \approx \\
     \approx \left(A_X\vec{Rg(B_0C_0^T)+t_0\mathbbm{1}^T} - b_X \right) + A_B\vec{\delta B} + A_{C^T}\vec{\delta C^T} + A_t\delta t
\end{multline}
where we now define
\begin{equation}
    r_{\mathrm{pOSE}} = A_X\vec{Rg(B_0C_0^T)+t_0\mathbbm{1}^T} - b_X
\end{equation}
with
\begin{equation}
    A_B = A_X(I \otimes R)\Gamma_g(C_0 \otimes I), \quad  A_{C^T} = A_X(I \otimes R)\Gamma_g(I \otimes B_0),\quad A_t = A_X(\mathbbm{1} \otimes I),
\end{equation}
where $\Gamma_g$ maps the elements from $BC^T$ to $g(BC^T)$ such that
\begin{equation}
    \vec{g(BC^T)} = \Gamma_g\vec{BC^T}.
\end{equation}
Since the weights $a$ are applied to the singular values of $BC^T$, the weighted nuclear norm terms can be written as \eqref{eq:sup_regB} and \eqref{eq:sup_regC}, similarly to the Low-rank Matrix Recovery problem. The residuals and jacobian in Algorithm 1 for the Non-rigid Strucuture Recovery problem can be computed as
\begin{equation}
    J = \begin{bmatrix} A_B & A_{C^T} & A_t\\
    A_{\mathrm{regB}} & \mathbf{0} & \mathbf{0}\\
    \mathbf{0} & A_{\mathrm{regC}} & \mathbf{0}
    \end{bmatrix}, \quad r_a = {\mathcal{A}}_a(B_0C_0^T) + b_a = \begin{bmatrix}
    r_{\mathrm{pOSE}}\\r_{\mathrm{regB}}\\r_{\mathrm{regC}}
    \end{bmatrix},
\end{equation}
and the translation is also added to the auxiliary variable $z$ in Algorithm 1, i.e., $z = [\vec{B};\vec{C^T};t]$.

\section{Results on Back, Heart and Paper Datasets}
In Figure~\ref{fig:back_heart_paper_results} we show an example of the reprojection errors obtained for the Back, Heart and Paper datasets in Section 5.2, for the weighted nuclear norm regularization and $\eta = 0.05$ (near perspective). Even though the qualitative difference between the methods is not visible (note the y-axis scale on the plots in Figure 1), the second-order method was still able to obtain a lower loss than all the first-order methods.

\section{Results on NRSfM Challange Datasets}
In this section we provide all results obtained with the weighted nuclear norm for the perspective camera model of the NRSfM Challange datasets. These include the the log-losses (Table~\ref{tab:log_loss}) and 3D reconstruction errors (Table~\ref{tab:rec_err}) for the ADMM and our method, in each of the six sequences (Circle, Flyby, Line, Semi-circle, Tricky, Zigzag) of the five datasets. Recall that the values in Tables 1 and 2 in the main text correspond to the average over the six sequences, for each dataset. In Figures~\ref{fig:nrsfm_1} and \ref{fig:nrsfm_2} we also show the qualitative comparison between the 3D reconstruction obtained with the two methods and the provided 3D ground-truth structure, for each sequence.

Note that our method is always able to obtain a lower loss compared to the ADMM, and the 3D reconstruction is always as good or much better (see the cases of Ballon-Semi-circle, Balloon-Tricky, Paper-Tricky, and Stretch-Flyby). The only exception was the sequence Tearing-Zigzag, where a lower loss actually resulted in a worse 3D reconstruction, which might be explained by incorrect modeling ($K = 2$ might be too low for this sequence).

\begin{figure}[t]
\centering
\begin{subfigure}{1\textwidth}
\includegraphics[height=1.745cm]{images/back_all_4.png}
\end{subfigure}
\begin{subfigure}{1\textwidth}
\includegraphics[height=1.85cm]{images/heart_all_4.png}
\end{subfigure}
\begin{subfigure}{1\textwidth}
\includegraphics[height=1.83cm]{images/paper_all_4.png}
\end{subfigure}
\caption{Comparison between reprojections (green) and 2D measurements (blue) obtained in Section 5.2 for Back (top), Heart (middle), and Paper (bottow) datasets. }
\setlength{\belowcaptionskip}{-40pt}
\label{fig:back_heart_paper_results}
\end{figure}
\begin{table}[t]
\setlength{\abovecaptionskip}{-1pt}
\caption{Log-loss on each for all sequences of the perspective datasets.}
\begin{tabular}{>{\centering}p{0.12\textwidth}>{\centering}p{0.2\textwidth}>{\centering}p{0.10\textwidth}>{\centering}p{0.10\textwidth}>{\centering}p{0.10\textwidth}>{\centering}p{0.10\textwidth}>{\centering}p{0.10\textwidth}>{\centering\arraybackslash}p{0.10\textwidth}}
  & \multicolumn{1}{c|}{Method \textbackslash Sequence}  & Circle & Flyby & Line & Semi-circle & Tricky & Zigzag \\ \cline{2-8}
\multicolumn{1}{c}{\multirow{2}{*}{Articulated}} & \multicolumn{1}{c|}{ADMM-WNN} & -1.822 & -2.849 & -3.797 & -3.405 &-3.009  & -2.517 \\
\multicolumn{1}{c}{} & \multicolumn{1}{c|}{Ours-WNN} & \textbf{-1.825} & \textbf{-2.853} & \textbf{-3.845} & \textbf{-3.408} & \textbf{-3.030} & \textbf{-2.753} \\ \hline \hline
\multirow{2}{*}{Balloon} & \multicolumn{1}{c|}{ADMM-WNN} & -2.232 & -2.977 & -3.130 & -2.607 &-2.380  &  -3.834\\
 & \multicolumn{1}{c|}{Ours-WNN}& \textbf{-2.465} & \textbf{-3.325} & \textbf{-3.096} & \textbf{-2.949} & \textbf{-2.934}  & \textbf{-4.037}\\\hline \hline \multirow{2}{*}{Paper} & \multicolumn{1}{c|}{ADMM-WNN} & -1.451 & -3.037 & -3.822 & -3.171  &-3.112  & -3.473 \\
 & \multicolumn{1}{c|}{Ours-WNN} & \textbf{-2.107 } & \textbf{-3.037} & \textbf{-3.823} & \textbf{-3.171} & \textbf{-3.809} & \textbf{-3.498}\\\hline \hline
 \multirow{2}{*}{Stretch} & \multicolumn{1}{c|}{ADMM-WNN} & -2.267 & -2.253 & -3.629 & -2.722 & -3.574 & -4.542 \\
 & \multicolumn{1}{c|}{Ours-WNN} &\textbf{-2.275}  & \textbf{-3.153} & \textbf{-3.846} & \textbf{-2.724} & \textbf{-3.578} &\textbf{-4.546}\\\hline \hline
 \multirow{2}{*}{Tearing} & \multicolumn{1}{c|}{ADMM-WNN} & -1.834 & -1.154 & -3.302 & -1.888 & -3.504  & -1.612 \\
 & \multicolumn{1}{c|}{Ours-WNN} & \textbf{-2.184} & \textbf{-1.662} & \textbf{-3.302} & \textbf{-2.067} & \textbf{-3.521} & \textbf{-2.017}
\end{tabular}
\label{tab:log_loss}
\end{table}
\begin{table}[h]
\setlength{\abovecaptionskip}{-1pt}
\caption{3D reconstruction error, in millimeters, on each for all sequences of the perspective datasets  relatively to the provided ground-truth structure.}
\begin{tabular}{>{\centering}p{0.12\textwidth}>{\centering}p{0.2\textwidth}>{\centering}p{0.10\textwidth}>{\centering}p{0.10\textwidth}>{\centering}p{0.10\textwidth}>{\centering}p{0.10\textwidth}>{\centering}p{0.10\textwidth}>{\centering\arraybackslash}p{0.10\textwidth}}
  & \multicolumn{1}{c|}{Method \textbackslash Sequence}  & Circle & Flyby & Line & Semi-circle & Tricky & Zigzag \\ \cline{2-8}
\multicolumn{1}{c}{\multirow{2}{*}{Articulated}} & \multicolumn{1}{c|}{ADMM-WNN} & 15.69 & \textbf{9.52} & 13.33 & 16.49 & \textbf{27.77} & 26.65 \\
\multicolumn{1}{c}{} & \multicolumn{1}{c|}{Ours-WNN} & \textbf{13.84} &  9.67 & \textbf{12.35}& \textbf{14.32} & 32.49 & \textbf{16.52}\\ \hline \hline
\multirow{2}{*}{Balloon} & \multicolumn{1}{c|}{ADMM-WNN} & 3.56 & \textbf{2.64} & \textbf{4.73}  & 16.06 & 24.46& 2.23 \\
 & \multicolumn{1}{c|}{Ours-WNN}& \textbf{2.07}  & 2.92 & 4.78 &\textbf{ 5.48} & \textbf{20.19}  & \textbf{2.19} \\\hline \hline \multirow{2}{*}{Paper} & \multicolumn{1}{c|}{ADMM-WNN} & 8.62 & \textbf{4.71} & \textbf{6.71} & 6.12 & 30.45 & \textbf{4.22} \\
 & \multicolumn{1}{c|}{Ours-WNN} & \textbf{1.98} & 4.75 & 7.06  & \textbf{6.02} & \textbf{9.80} & 4.45 \\\hline \hline
 \multirow{2}{*}{Stretch} & \multicolumn{1}{c|}{ADMM-WNN} & \textbf{2.59} & 16.86 & \textbf{4.78} & 5.68 & 15.65 & \textbf{2.80}  \\
 & \multicolumn{1}{c|}{Ours-WNN} & 2.69 & \textbf{2.85} & 6.74  & \textbf{5.58} & \textbf{14.93} & 2.84 \\\hline \hline
 \multirow{2}{*}{Tearing} & \multicolumn{1}{c|}{ADMM-WNN} & 5.10 & 10.94 & \textbf{8.93} & 5.05 &  18.57 & \textbf{7.09} \\
 & \multicolumn{1}{c|}{Ours-WNN} & \textbf{4.25}  & \textbf{7.15} &  9.17 & \textbf{4.87}& \textbf{16.98}  & 8.12
\end{tabular}
\label{tab:rec_err}
\end{table}

\newpage
\iffalse
\begin{figure}
\begin{subfigure}{0.01\textwidth}
\rotatebox[origin=t]{90}{Circle}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[height=2cm]{images/articulated_circle_3D.png}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[height=2cm]{images/articulated_flyby_3D.png}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[height=2cm]{images/articulated_line_3D.png}
\end{subfigure}

\begin{subfigure}{0.19\textwidth}
\includegraphics[height=2cm]{images/articulated_semi_circle_3D.png}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[height=2cm]{images/articulated_tricky_3D.png}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[height=2cm]{images/articulated_zigzag_3D.png}
\end{subfigure}
\end{figure}
\fi

\begin{sidewaysfigure}
\begin{subfigure}{1\textwidth}
\hspace{1.5cm}Circle Sequence
\hspace{5.5cm}Flyby Sequence
\hspace{5cm}Line Sequence
\end{subfigure}
\vspace{0.1cm}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/articulated_circle_3D.png}
\hspace{3cm}
\includegraphics[height=2.25cm]{images/articulated_flyby_3D.png}
\hspace{2.5cm}
\includegraphics[height=2.25cm]{images/articulated_line_3D.png}
\end{subfigure}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/balloon_circle_3D.png}
\hspace{2.5cm}
\includegraphics[height=2.25cm]{images/balloon_flyby_3D.png}
\hspace{2.15cm}
\includegraphics[height=2.25cm]{images/balloon_line_3D.png}
\end{subfigure}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/paper_circle_3D.png}
\hspace{2.5cm}
\includegraphics[height=2.25cm]{images/paper_flyby_3D.png}
\hspace{2cm}
\includegraphics[height=2.25cm]{images/paper_line_3D.png}
\end{subfigure}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/stretch_circle_3D.png}
\hspace{2.7cm}
\includegraphics[height=2.25cm]{images/stretch_flyby_3D.png}
\hspace{2cm}
\includegraphics[height=2.25cm]{images/stretch_line_3D.png}
\end{subfigure}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/tearing_circle_3D.png}
\hspace{2.7cm}
\includegraphics[height=2.25cm]{images/tearing_flyby_3D.png}
\hspace{2cm}
\includegraphics[height=2.25cm]{images/tearing_line_3D.png}
\end{subfigure}
\caption{Comparison between the estimated (blue) and provided (red) 3D structure for the sequences Circle, Flyby and Line.}
\label{fig:nrsfm_1}
\end{sidewaysfigure}

\begin{sidewaysfigure}
\begin{subfigure}{1\textwidth}
\hspace{1cm}Semi-circle Sequence
\hspace{5.25cm}Tricky Sequence
\hspace{5cm}Zigzag Sequence
\end{subfigure}
\vspace{0.1cm}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/articulated_semi_circle_3D.png}
\hspace{3cm}
\includegraphics[height=2.25cm]{images/articulated_tricky_3D.png}
\hspace{2.5cm}
\includegraphics[height=2.25cm]{images/articulated_zigzag_3D.png}
\end{subfigure}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/balloon_semi_circle_3D.png}
\hspace{2.5cm}
\includegraphics[height=2.25cm]{images/balloon_tricky_3D.png}
\hspace{2.15cm}
\includegraphics[height=2.25cm]{images/balloon_zigzag_3D.png}
\end{subfigure}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/paper_semi_circle_3D.png}
\hspace{2.5cm}
\includegraphics[height=2.25cm]{images/paper_tricky_3D.png}
\hspace{2cm}
\includegraphics[height=2.25cm]{images/paper_zigzag_3D.png}
\end{subfigure}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/stretch_semi_circle_3D.png}
\hspace{2.7cm}
\includegraphics[height=2.25cm]{images/stretch_tricky_3D.png}
\hspace{2cm}
\includegraphics[height=2.25cm]{images/stretch_zigzag_3D.png}
\end{subfigure}

\begin{subfigure}{1\textwidth}
\includegraphics[height=2.25cm]{images/tearing_semi_circle_3D.png}
\hspace{2.7cm}
\includegraphics[height=2.25cm]{images/tearing_tricky_3D.png}
\hspace{2cm}
\includegraphics[height=2.25cm]{images/tearing_zigzag_3D.png}
\end{subfigure}
\caption{Comparison between the estimated (blue) and provided (red) 3D structure for the sequences Semi-circle, Tricky and Zigzag.}
\label{fig:nrsfm_2}
\end{sidewaysfigure}

redpajama_set_name: RedPajamaArXiv


text:
Metal Band Rammstein Kisses Onstage to Protest Russian Anti-LGBTQ Laws
Jael Goldfine
Rammstein is a freaky, cult German metal band, best known for storming their shows in BDSM costumes with flamethrowers, that I learned about in a college class about punk rock. Extremely lame, yes.
We watched a performance of the six-piece during which their hyper-masculine, mountainous frontman Till Lindeman squirted the crowd with a mysterious liquid from a swinging strap-on. We talked about queerness and performance, and compared how Lindeman's antics would be read in comparison to those of Lynn Breedlove, the swaggering lesbian frontwoman of queercore punk act Tribe8, who also frequently performed wearing a dildo.
With this as my only context, it was very cool and not entirely surprising to see this photo of Rammstein from their recent concert in Moscow. Guitarists Paul Landers and Richard Kruspe ended the show by making out on-stage, and saying "Russia, we love you," flying in the face of the Russian government's crackdown on LGBTQ people.
Rammstein on Instagram: "Россия, мы любим тебя! Photos: @jenskochphoto"
Their steamy performance tactic is specifically being linked to a policy called the "gay propaganda" law, that bars "the promotion of non-traditional sexual relationships" to minors, which based on its usage so far, includes films like Elton John biopic Rocketman and pretty much all LGBTQ rights activism. It can even get foreigners like Rammstein "arrested and detained for up to 15 days, then deported, or fined up to 5,000 rubles and deported."
Related | Activists Arrested for Protesting Chechnya's Abuse of LGBTQ People
The law has also justified the lack of judicial action taken against hate crimes targeting LGBTQ people. As recently as last week, LGBTQ activist Yelena Grigoryeva was stabbed and killed after her information was shared on an anti-LGBTQ vigilante Facebook group, which Russian authorities have reportedly failed to investigate.
According to The AV Club, the Rammstein kiss wasn't a totally new addition to their set, and thus not entirely about protesting Russian policies. Apparently, the band has been ending their shows with a make out session throughout their current European tour, which, given the band is all straight, as far as we know, is perhaps a questionable shock tactic.
Related | Russian Official Threatens LGBTQ Leader: "Go Back to Where You Belong"
However, huge, scary looking dudes sucking face front of thousands of people in country full of rampant, institutionalized homophobia — and getting people talking about the oppression of queer Russians — you have to love it. Rammstein's camp and leather has always had a queer sensibility, and this might just bring to icon status.
Photo via Instagram
Marti Cummings Could Be New York's First Nonbinary City Councilor
Ezra Furman Is Angry. Aren't You?
Sandra Song

redpajama_set_name: RedPajamaCommonCrawl


text:
Pharmacy Benefit
Ancillary Benefits
Technological Excellence
2020 WORKERS' COMP
INDUSTRY INSIGHTS SURVEY
Delivered by Healthesystems RxInformer
Powered by Risk & Insurance
TOP CHALLENGES
PROGRAM CHALLENGES
POPULATION TRENDS
TECHNOLOGICAL ADVANCES
CLAIMS RISKS
CURRENT OPINION
Past Surveys
Healthesystems embarked on its second annual collaboration with Risk & Insurance magazine to survey workers' compensation industry stakeholders about issues regarding the provision of medical care for injured workers. Our goal was to understand their current concerns and draw insights that will help inform the many crucial decisions that our customers and colleagues in workers' comp make every day to ensure the best possible care for injured workers.
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Conducted in-person at NWCDC 2019 and via online through Risk & Insurance, the 2020 Workers' Comp Industry Insights Survey drew responses from 669 industry stakeholders, all of whom first identified their organization type and individual role. This allowed us to analyze results collectively, while also examining the ways in which views sometimes varied between industry segments and professions.
70% report changing population is top industry challenge
Three issues stood out as being top-of-mind for the vast majority of participants: the changing workforce population; comorbidities/poor worker health; and the increasing complexity of claims. With some minor exceptions, these were considered the top three challenges for our industry, an assessment that appears all the more accurate as the COVID-19 virus strikes. And we see clear connections between these challenges. For example, older workers, the most impactful population trend by a wide margin, are more likely to suffer comorbidities, which are a leading contributor to medical complications and complex medical claims.
Growing concern about the ability to manage claims was revealed in multiple incidences throughout the survey. For example, claims process/workflow automation was consistently selected as the #2 most important technical advancement over the next 3 – 5 years (telemedicine was #1). And over 80% of participants agreed that the greater complexity of claims is requiring more clinical decisions to be made during the claims management process. Those decisions put additional pressure on claims professionals and case managers already under heavy workloads, increasing the risk that something will be missed.
83% report complex claims mean more clinical decisions for claims management
Arguably the biggest (pre-COVID-19) risk for workers' comp programs over the past decade has been opioid abuse, but concern about opioids appears to be on the decline. Opioid abuse did not make it into the list of top 5 industry challenges in this year's survey (as compared to #3 last year), nor did participants see it as a top health or claims risk. As an industry, we have made progress in curbing opioid abuse, but we still have work to do and it would be premature to claim victory. A view apparently shared by insurance carriers who bucked the trend by ranking opioids/substance abuse their #1 claims risk.
PTSD and other mental health conditions have surpassed opioids as a perceived challenge and risk. This is probably due to a combination of two factors: an increasing number of states expanding coverage for first responders and an overall increase in reported incidences of mental health conditions, driven in part by the large number of millennials now in the workforce who are more likely to experience and seek treatment for conditions such as anxiety and depression.
PTSD/mental health conditions surpassed opioids as perceived risk
Escalating medical costs was the #1 challenge for survey respondents in last year's survey, so we dug a little deeper this year and learned that containing hospital costs is the top priority overall, followed by Rx drugs and physician/professional fees. Responses to the cost question tended to vary, however, depending on the type of organization or individual role. For example, insurance carriers were more focused on Rx drug costs than hospital costs and participants in clinical roles put physical medicine ahead of physician/professional fees.
While the overall trends revealed in the survey were consistent, results varied on some questions, depending on who was responding. For example, executive leaders were more concerned about legislative and regulatory changes than other participants, ranking it as their second most important challenge. Executive leaders also have more confidence in artificial intelligence as an important technology, ranking as the second most important technology over the next 3-5 years, as opposed to other participants for whom artificial intelligence was not seen as important. Information technology and/or data analytics professionals were not among the survey participants and it is possible that executive leaders know more about how artificial intelligence is being applied in workers' comp.
Medical program managers ranked opioids and substance abuse as their number one program challenge, in contrast to number four ranking opioids and substance abused received overall. This is likely because medical directors and managers are keenly aware that opioid use is still prevalent and risky. Medical program managers were also more concerned than any other group about new and expensive medical treatments, possibly because they have greater insight into the efficacy and cost-benefit ratios of new drugs, devices, and therapies.
Medical program managers still see opioids as top challenge
The type of organization participants worked for also seems to have influenced their perspectives with insurance carrier participants expressing greater concern about opioids and substance abuse, as well as prescription drug prices, as compared to those from employers and government agencies. And claims professionals and case managers stood out in their selection of comorbidities as the number one claims risk, which came in at number three in the overall results. These variations may reflect their closer familiarity with claims and the factors that can drive them off course.
The aging workforce and increased prevalence of comorbidities, among both older and younger workers, are contributing to more complex claims, and now more serious complications for COVID-19 patients (almost 90% of COVID-19 patients admitted to the hospital suffer from comorbidities). And the aging workforce presents other challenges for workers' comp. For example, the average worker age in the insurance and healthcare industries is also rising, and a wave of retirements will likely lead to a shortage of experienced people to process claims and care for injured and ill workers, just as demand is increasing.
Technology will help to bridge the gaps. Increased use of telemedicine in workers' comp will help alleviate long wait times to see physicians and should also make those visits less expensive. Telemedicine has also proved to be an invaluable tool for providing access to care without risk of infection during the current pandemic.
Improved claims automation will allow claims professionals to focus their time and attention where it is most needed, help guide clinical decisions, and improve communications between the claims and care management teams. Enhanced data visualization and predictive analytics will be crucial in detecting and mitigating risks early in the claims process. And we should expect new risks, not only from dramatic events like the COVID-19 pandemic, but also from more commonplace occurrences.
We know from past experience that changes are often accompanied by unexpected consequences. New drugs to treat chronic pain brought the opioid crisis, and the current movement away from opioids could lead to other forms of inappropriate prescribing. Treating comorbidities may result in polypharmacy and the risk of adverse drug reactions. Medical innovations, such as biologics and new medical devices, present new opportunities for promising treatment alternatives, but also for fraud, waste, and abuse. And legislative changes, such as expanding coverage for PTSD and other mental health conditions will help many patients receive much-needed treatment but might also increase caseloads and costs.
The 2020 Workers' Comp Industry Insights Survey shows us that workers' comp professionals are keenly aware of the wide-scale changes taking place. Our job as industry leaders is, whenever possible, to anticipate the implications of those changes and prepare for them.
report changing population is a top industry challenge
report complex claims mean more clinical decisions for claims management
report better claims efficiency will lower medical costs
report top claims risk is the failure to detect early warning signs before they escalate
Overview of Participants
669 industry stakeholders responded to the survey:
PARTICIPANT ORGANIZATIONS
Employer 214
Brokerage 92
Insurance Carrier 88
Healthcare Provider 69
State/Government Agency 54
Consultancy 50
Third Party Administrator 30
Law Firm 26
Managed Care Organization 23
to Reveal Organizations
and Roles
PARTICIPANT INDIVIDUAL ROLES
154 Claims Management
141 Executive Leadership
120 Risk Management
89 Broker/Agent
37 Legal/Regulatory
33 Clinical/Case Management
28 Healthcare Provider
16 Medical Program Management
8 Procurement
43 Other Roles
See more on job role perspectives >>
Top Industry Challenges
The top 5 biggest challenges facing the workers' compensation industry in 2020
The changing workforce/population demographics
Increasing complexity of claims
Comorbidities and poor worker health
New and expensive medical treatments
Regulatory and legislative changes
Executive leaders ranked regulatory and legislative changes more highly than any other group, ranking it as their second most important challenge.
COVID-19 Consideration
People ages 50-59 are 3X more likely to die from the COVID-19 virus and those aged 60-69 are 10X more likely to die than younger counterparts.1
The top 5 challenges for participants' workers' comp programs in 2020
Changing workforce population
Opioids and substance abuse
Mental health exposures
All roles were in agreement on the top 3 program challenges EXCEPT medical program managers whose top 3 challenges were:
1. Opioid and substance abuse
2. New and expensive medical treatments
3. Mental health exposure/PTSD coverage
Comorbidities and poor health are a serious concern for COVID-19 patients with 19% of patients with at least one comorbidity and 29% of patients with two or more comorbidities being admitted to intensive care, as compared with 8% of patients overall. 2
Population trends that will have the biggest impact on workers' comp programs in 2020
Millennial/younger worker influence
Temporary/gig worker trend
First responder coverage
Racial, ethnic and gender diversity
The aging workforce was the #1 population challenge across all organizations and roles.
Nearly ¾ of workers aged 65 or older - 5 million people - cannot work remotely due to the nature of their jobs and extra precautions may be needed to protect the health of these workers during the pandemic.3
Most important technologies over the next 3-5 years
Claims processing/workflow automation
Data/predictive analytics
These were the top 3 technologies selected by most roles with the exception of executive leaders who ranked Artificial Intelligence as the 2nd most important technology.
Since the virus began spreading in the US, at least 18 states have adjusted their worker's comp rules to allow for increased access to telehealth for injured workers.4
Most concerning health risks within claims populations
Mental health/PTSD
Opioids/Substance Abuse
Respondents across the board ranked chronic pain their top concern and almost put mental health/PTSD ahead of opioids and substance abuse, which could indicate a shift in workers' comp medical management priorities.
Some chronic conditions appear to pose more risk than others for COVID-19 patients, with one large study reporting that of patients hospitalized 57% had hypertension, 41% were obese, and one third had diabetes.5
Medical cost containment priorities in 2020
Rx Drug Costs
Physician/professional fees
DME, diagnostics, and other ancillary benefits
Insurance carriers differed slightly with overall results, ranking Rx drugs costs as their #1 area of focus and hospital costs #2.
Hospital costs for COVID-19 patients are estimated to be in the $10,000 - $20,000 range if no ventilator treatment is required. With ventilator treatment, costs can average nearly $90,000.6
Most concerning claims risks
Not detecting claim warning signs early enough
Charges for medical services unrelated to workplace injury
Undetected fraud
Injured worker opioid/substance abuse
Lack of transparency into medical prices
New medical treatments and drugs
Insurance carriers ranked injured worker opioid substance abuse as their #1 risk. Claims managers and clinical/case managers selected comorbidities as the #1 risk.
Some states have explicitly given WC coverage to certain workers,4 and many, along with the federal government, have mandated group health coverage for the virus.7 For "essential" workers who are not in public safety and healthcare jobs, compensability remains a big question.
Most to least agreed statements on current topics
The increasing complexity of care is requiring more clinical decisions to be made during the claims management process
Improved claims efficiency can help lower medical costs
The aging healthcare workforce and anticipated physician shortage will drive up medical costs in workers' comp
Prescription drug activity is an early and reliable indicator of potential risks
Medical marijuana will be covered by workers' comp insurance in more state over next 2-5 years
Medical advance and new treatment present new opportunities for FWA
The cost saving potential from ancillary benefits (i.e. DME, Transportation) is underestimated by many workers' comp payers (only statement that had more neutral and disagree than agree)
Claims managers most agreed that improved claims efficiency can help lower medical costs, while risk managers most agreed that the anticipated physician shortage will drive up medical costs in workers' comp.
Although workers' comp claims may decrease temporarily due to business closures and telecommuting, the frequency and severity of workers' comp claims is expected to increase in the wake of the COVID-19 pandemic,8 demanding even more decisions regarding an entirely new disease.
1. Kofman, A. and Hernandez-Romieu, A. "Protect older and vulnerable healthcare workers from COVID-19." STAT. March 25, 2020. https://www.statnews.com/2020/03/25/protect-older-and-vulnerable-health-care-workers-from-covid-19/
2. Glick, T. et. al. "Association of Comorbidities with COVID-19 Outcomes." New England Journal of Medicine, Journal Watch. April 1, 2020. https://www.jwatch.org/na51250/2020/04/01/association-comorbidities-with-covid-19-outcomes
3. U.S. Bureau of Labor Statistics. Job Flexibilities and Work Schedules. September 24, 2019. https://www.bls.gov/news.release/pdf/flex2.pdf
4. Healthesystems. "States Move to Make COVID-19 Compensable." April 20, 2020. https://medmonitor.healthesystems.com/regulatory-developments/states-move-to-make-covid-19-compensable/
5. Richardson, S. et.al. "Presenting Characteristics, Comorbidities, and Outcomes Among 5700 Patients Hospitalized with COVID-19 in the New York City Area." Journal of the American Medical Association. April 22, 2020. https://jamanetwork.com/journals/jama/fullarticle/2765184?guestAccessKey=906e474e-0b94-4e0e-8eaa-606ddf0224f5&utm_source=For_The_Media&utm_medium=referral&utm_campaign=ftm_links&utm_content=tfl&utm_term=042220
6. Rae M. et. al. "Potential Costs for COVID-19 Patients with Insurance Coverage." Peterson-KFF Healthsystem Tracker. March 13, 2020. https://www.healthsystemtracker.org/brief/potential-costs-of-coronavirus-treatment-for-people-with-employer-coverage/
7. Pollitz, K. "Private Health Coverage of COVID-19: Facts and Issues." Kaiser Family Foundation. March 18, 2020. https://www.kff.org/private-insurance/issue-brief/private-health-coverage-of-covid-19-key-facts-and-issues/
8. Hussein, F. "Workers' Comp Premiums Could Skyrocket with COVID-19 Claims." Bloomberg Law. March 30, 2020. https://news.bloomberglaw.com/daily-labor-report/workers-comp-premiums-could-skyrocket-with-covid-19-claims
info@healthesystems.com
©2019 Healthesystems


Vegetables need to be grown evenly spaced in straight rows surrounded by bare soil. Some gardeners feel the need to lavish them with chemicals to make them grow bigger and spray them with yet more chemicals to stop pests devouring them.
Here, at Nant-y-bedd garden we don't do any of the above and yet we still managed to win the First Prize and Challenge cup at the local Llanthony Show for the Most Productive Vegetable Garden for the second year running.
There are a few basic rules that apply to any system of growing veg organically and we follow these: get to know your soil and growing conditions and look after your soil; rotate crops to reduce the risk of pests and diseases affecting your crops; use organically-approved methods of dealing with those pests and diseases. None of this is particularly earth-shatteringly different. Any gardener keen to produce tasty, fresh veg with the minimum of artificial inputs would be doing the same.
However, some of what we do here challenges devoutly-held perceptions of what constitutes good gardening practice.
Top of the list is that we 'embrace our weeds'. We do this for a number of reasons and in a variety of ways. Ground elder, Hogweed and Rose Bay Willowherb, for example, are undoubtedly not what you want in your veg beds because they are invasive if not controlled but the insect pollinators on which we depend love them, so we allow them to have their place around the veg beds.
We don't weed out self-sown seedlings just because we didn't sow them there. If they are not compromising the crop they can stay because the bees like them/they look pretty/they confuse the pests looking for the crops/we eat them – any or all of these reasons apply.
We save our own seed thereby over time developing our own strains of veg which are best adapted to our growing conditions rather than buying seed from a supplier where the seed may have come from an area with very different growing conditions.
We have other favourite ways of foiling pests – for example, ducks and copper for slug control and netting and collars to protect brassicas.
As well as saving our own seeds, each year we try different Heritage varieties to seek out those which might do well here. We belong to the Heritage Seed Library. The greatest success this year has been discovering the Climbing French Beab 'Blue Coco' which we have been picking all summer. We will save the seed and grow it again but of course the weather this year has been exceptional – perfect even here at 1200 feet in the Black Mountains for growing French beans – next summer might be very different!
Ground elder flowering. Did you know that you can eat it?
Oh yes, and we are growing our veg the 'no-dig' way these days – putting a layer of compost on top of the soil to avoid disturbing the magical networks which operate in the top layers of the soil about which we know so little. Mimicking, in effect, how 'nature' does it.
Perhaps I'll add a new workshop to the Nant-y-bedd Programme for 2019 – 'Introduction to veg growing Nant-y-bedd style'? Let us know if you would be interested.


Client: Vodokanal of St. Petersburg is an official water supplier of St. Petersburg. It provides 4.5 million citizens with drinking water and sewage services. Vodokanal is one of national leaders in innovations. Company is applying a major investment program aimed to reduce quantity of wastewater and improve quality of its treatment. Vodokanal is 293rd largest Russian company, according to Finance magazine of Russia (2010).
Agency's main goal was to raise awareness of Clean Baltic international project aimed to stop dumping of non-treated wastewater to water bodies of St. Petersburg. To accomplish it, we have conducted an online ceremony of connection of direct wastewater outlets to the Main Sewage Collector and forced extensive media coverage of the event.
The ceremony was visited by the Governor of St. Petersburg, Consul General of Finland in Russian Federation, representatives of public organizations and governing body of Vodokanal. We have also developed a presentation film with 3D animation of the Main Collector and its most important parts.


When you have over 100 Christmas trees you become a self appointed pro of holiday storage. You guys have seen how I do Halloween, Easter even St. Patrick's Day – the key to being a holiday hoarder is organization. That and LOT'S of storage bags and bins. I mean LOT'S of them.
This is a sponsored post for Treetopia (I am their paid brand ambassador) let's be clear on that right up front. However, I was a holiday junkie long before that and I've learned a few tips, tricks and hacks in my years all about holiday storage. I had to or my closets would be popping open with Christmas trees. OK, two closets in my house are popping open with trees, but today I'm talking about my attic.
So how do you store your holiday decorations? Are you an attic, closet or under the bed type? Got any fun tips to share? I'm always eager to learn new ideas because lord knows I'm always collecting more holiday goodies!
Posted January 4, 2018 & filed under Christmas, Holidays.


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Depression? She'll get over it! Off work because of stress? He's having a laugh! Funny isn't it?
Our emotional health is a serious issue, but we still don't take it seriously. We avoid it. We misunderstand it. We even joke about it. We do everything, really, except talk about it. So how serious is the issue?
One in four of us suffer from depression at some time in our lives
Over five million working days are lost each year due to stress
Last year more people died in Britain from suicides than car accidents.
It's that serious it affects us all
SAMARITANS were founded in November 1953 when Chad Varah, Vicar of St Paul's, Clapham Junction, London recognised that there was a need for an emergency number for those who were feeling suicidal, just as there was for other emergencies. A group of volunteers manned telephones, and aimed to 'befriend' callers who were suicidal. By 1966, there were 6,537 Samaritan volunteers based in 80 Branches.
Calls to Samaritans continued to increase steadily, and today Samaritans is the UK and Republic of Ireland 's leading help line, with 203 Branches taking 4.8 million calls from people in distress each year. 18,700 volunteers man our telephone and email contact lines nationally. Samaritans offers callers the time and space to talk about their problems and to explore their feelings enabling them to be better able to cope with their lives.
CHELTENHAM AND DISTRICT SAMARITANS BRANCH was founded in 1963, offering support to those in distress in Cheltenham itself, and in the outlying rural areas. In 2003-4 the Branch had over 37,000 contacts, including over 16,000 telephone calls. In June 1994, Samaritans Cheltenham became the first Branch to offer completely confidential emotional support via an email service. Since this first pilot scheme, more and more Samaritans branches have started offering this service. In 2003-4 Cheltenham and District Branch responded to nearly 1,700 emails from all over the country.
Samaritans is not just a help line service, and funds enable us to reach more people with targeted projects via our Outreach programme. For example, Cheltenham and District Samaritans volunteers work closely with both prisoners specially trained as Listeners, and with staff at Long Lartin Prison. We also work regularly with schools, and in 2004 we once again continued to fulfil a variety of speaking engagements.
Help us to support the vital work of Cheltenham and District Branch of Samaritans


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Beloved, zany coach and legendary broadcaster John Madden dies
December 29, 2021 by Paul Walker
John Madden retired from coaching at age 42 and had a 30-year career as a broadcaster.
Image: Getty Images
Famed Raiders' coach and NFL analyst John Madden died unexpectedly Tuesday morning at the age of 85.
In an official statement from the NFL, commissioner Roger Goodell extended his condolences to Madden's family and gave honor to the Hall of Famer's storied career:
We all know him as the Hall of Fame coach of the Oakland Raiders and broadcaster who worked for every major network, but more than anything, he was a devoted husband, father and grandfather.
Nobody loved football more than Coach. He was football. He was an incredible sound board to me and so many others. There will never be another John Madden, and we will forever be indebted to him for all he did to make football and the NFL what it is today.
Madden's impact on football was remarkable to say the least. When first brought on as head coach of the Oakland Raiders in 1969, Madden was the youngest coach in the AFL. Still, he led his Raiders to the best record (12-1-1) across both the AFL and NFL. He would go on to win a Super Bowl with the team in 1976 — a 32-14 victory over the Minnesota Vikings.
Madden retired as the Raiders' winningest coach and still holds that distinction to this day. His .759 win percentage is the highest ever for an NFL coach with at least 100 games. Only Guy Chamberlain has a higher all-time win percentage (.784). He was elected to the Pro Football Hall of Fame in 2006. Moreover, the disheveled, anti-corporate coach was a rebel and free thinker — the opposite of staid men like rival Tom Landry — who was a perfect match for owner Al Davis' renegade style.
But Madden left and even bigger legacy in his next career. After retiring from coaching, he continued to expand the appeal of football from the broadcast booth. Madden was well-known for his insightful takes and colorful commentary in a 30-year career that ended in 2008. that helped mainstream audiences across America understand the X's and O's of football that much better.
His ever-recognizable exclamations of "Boom!" or his signature too obvious sayings that sounded like great wisdom:
"I always used to tell that we are here to win. And you know what, Al? When you win, you don't lose!"
"If the quarterback throws the ball in the end zone and the wide receiver catches it, it's a touchdown."
Madden also became the face of the enormously popular EA Sports video game franchise of the same name that featured many of his Maddenisms. The yearly release has become a staple in any sports gamer's library and is the greatest simulation-style football franchise of all-time.
The bombastic Madden was an active developer in the early years of the game and aimed to create a game that accurately depicted the sport and the NFL as a league.
Madden spent the final years of his life serving as an advisor to the NFL, advocating for rule changes to help improve player safety.
Just this past week, Fox Sports aired a documentary paying homage to the legend's influence on the sport. The documentary will be available to stream on a variety of services including ESPN+ and Peacock beginning January 3.
Madden is survived by his wife of 62 years, Virginia, and two sons, Joseph and Michael.
Categories Sports Tags Al Davis, American football, American Football League, Deadspin, Electronic Arts, Electronic Arts games, football, Guy Chamberlain, John Madden, Joseph, Madadhan, Madden, Madden NFL, Michael, NFL, Pro Football Hall of Fame inductees, Roger Goodell, Sports, Super Bowl, the NFL, Tom Post navigation
John Madden dead at 85: Sports world reacts to death of legendary NFL coach, broadcaster
Former U.S. Senate majority leader Harry Reid dead at 82 | CBC News