text:
\section{Introduction}\label{sec:int}

Generative models, a growing area of unsupervised learning, aim to model the data distribution $p({\bm{x}})$ over data points ${\bm{x}}$ from a space $\mc{X}$, which is usually a high-dimensional Euclidean space $\mathbb{R}^n$. This has desirable benefits like a naturally definable inner-product,
vector addition, or a closed-form distance function.
Yet, many types of data have a strongly non-Euclidean latent structure \citep{noneucldata},
e.g.~the set of human-interpretable images.
They are usually thought to live on a ``natural image manifold''
\citep{natural_manifold}, a continuous lower-dimensional subset of the space in which they are represented. By moving along the manifold, one can continuously change the content and appearance of interpretable images.
As noted in \citet{hypemb},
changing the geometry of the underlying latent space enables better representations of specific data types compared to Euclidean spaces of any dimensions, e.g.~tree structures and scale-free networks.

Motivated by these observations, a range of recent methods learn representations
in different spaces of constant curvatures: spherical or elliptical \citep{sphemb},
hyperbolic \citep{hypemb,sala_hyp,hypglove}
and even in products of these spaces \citep{products,ccgcn}.
Using a combination of different constant curvature spaces, \citet{products} aim to match the underlying geometry of the data better.
However, an open question remains: how to choose the dimensionality and curvatures of each of the partial spaces?

A popular approach to generative modeling is the Variational Autoencoder \citep{vae}. VAEs provide a way to sidestep
the intractability of marginalizing a joint probability model
of the input and latent space $p({\bm{x}}, {\bm{z}})$,
while allowing for a prior $p({\bm{z}})$ on the latent space.
Recently,
variants of the VAE have been introduced for spherical \citep{svae, svae2}
and hyperbolic \citep{hvae, phg} latent spaces.

Our approach, the Mixed-curvature Variational Autoencoder, is a generalization of VAEs
to products of constant curvature spaces.\hspace{-0.3em}\footnote{Code is available on GitHub at \url{https://github.com/oskopek/mvae}.}
It has the advantage of a better reduction in dimensionality,
while maintaining efficient optimization.
The resulting latent space is a ``non-constantly'' curved manifold that is more flexible than a single constant curvature manifold.

Our contributions are the following:
(i) we develop a principled framework for manipulating representations and modeling probability distributions in products of constant curvature spaces that smoothly transitions across curvatures of different signs,
(ii) we generalize Variational Autoencoders to learn latent representations on products of constant curvature spaces with generalized Gaussian-like priors, and
(iii) empirically, our models outperform current benchmarks on a synthetic tree dataset \citep{hvae}
and on image reconstruction on the MNIST \citep{mnist}, Omniglot \citep{omniglot}, and CIFAR \citep{cifar} datasets for some latent space dimensions.

\section{Geometry and Probability in Riemannian manifolds}\label{sec:geom}

To define constantly curved spaces, we first need to define the notion of \emph{sectional curvature} $K(\tau_{\bm{x}})$ of two linearly independent vectors in the tangent space at a point ${\bm{x}} \in \mc{M}$ spanning a two-dimensional plane $\tau_{\bm{x}}$ \citep{panoramic_geo}. Since we deal with constant curvature spaces where all the sectional curvatures are equal, we denote a manifold's curvature as $K$.
Instead of curvature $K$, we sometimes use the generalized notion of a \emph{radius}: $R = 1 / \sqrt{|K|}.$

There are three different types of manifolds $\mc{M}$ we can define with respect to the sign of the curvature: a positively curved space, a ``flat'' space, and a negatively curved space. Common realizations of those manifolds are the hypersphere $\sph_K$, the Euclidean space $\euc$, and the hyperboloid $\hyp_K$:
\begin{align*}\mc{M} =
  \begin{cases}
    \sph^n_K = \{{\bm{x}} \in \mathbb{R}^{n+1} : \scprod{{\bm{x}}, {\bm{x}}} = 1/K\}, & \text{for } K > 0 \\
    \euc^n = \mathbb{R}^n, & \text{for } K = 0\\
    \hyp^n_K = \{{\bm{x}} \in \mathbb{R}^{n+1} : \lprod{{\bm{x}}, {\bm{x}}} = 1/K\}, & \text{for } K < 0
  \end{cases}\end{align*}
where $\scprod{\cdot, \cdot}$ is the standard Euclidean inner product,
and $\lprod{\cdot, \cdot}$ is the Lorentz inner product,
\begin{align*}\lprod{{\bm{x}}, {\bm{y}}} = -x_1 y_1 + \sum_{i=2}^{n+1} x_i y_i \quad  \forall {\bm{x}}, {\bm{y}} \in \mathbb{R}^{n+1}.\end{align*}

We will need to define the exponential map, logarithmic map, and parallel transport in all spaces we consider.
The exponential map in Euclidean space is defined as
$\exp_{\bm{x}}({\bm{v}}) = {\bm{x}} + {\bm{v}},$
for all ${\bm{x}} \in \euc^n$ and ${\bm{v}} \in \tang{{\bm{x}}}\euc^n$.
Its inverse, the logarithmic map is
$\log_{\bm{x}}({\bm{y}}) = {\bm{y}} - {\bm{x}},$
for all ${\bm{x}}, {\bm{y}} \in \euc^n$.
Parallel transport in Euclidean space is simply an identity
$\PT_{{\bm{x}} \to {\bm{y}}}({\bm{v}}) = {\bm{v}},$ for all ${\bm{x}}, {\bm{y}} \in \euc^n$ and ${\bm{v}} \in \tang{{\bm{x}}}\euc^n$.
An overview of these operations in the hyperboloid $\hyp_K^n$ and the hypersphere $\sph_K^n$ can be found in Table~\ref{tab:summary_ops}.
For more details, refer to \citet{petersen}, \citet{cannon}, or Appendix~\ref{app:geom}.

\subsection{Stereographically projected spaces}

The above spaces are enough to cover any possible value of the curvature,
and they define all the necessary operations we will need to train VAEs in them.
However, both the hypersphere and the hyperboloid have an unsuitable property, namely the non-convergence of the norm of points as the curvature goes to 0.
Both spaces grow as $K \to 0$ and become locally ``flatter'', but to do that, their points have to go away from the origin of the coordinate space $\B{0}$ to be able to satisfy the manifold's definition.
A good example of a point that diverges is the origin of the hyperboloid (or a pole of the hypersphere)
${\bm{\mu}}_0 = (1/\sqrt{|K|}, 0, \ldots, 0)^T$. In general, we can see that $\norm{{\bm{\mu}}_0}^2 = 1/K \xrightarrow{K \to 0} \pm\infty$.
Additionally, the distance and the metric tensors of these spaces do not converge to their Euclidean variants as $K \to 0$, hence the spaces themselves do not converge to $\mathbb{R}^d$.
This makes both of these spaces unsuitable for trying to learn sign-agnostic curvatures.

Luckily, there exist well-defined non-Euclidean spaces that inherit most properties
from the hyperboloid and the hypersphere, yet do not have these properties -- namely,
the Poincar\'{e} ball and the projected sphere, respectively.
We obtain them by applying stereographic conformal projections, meaning that angles are preserved by this transformation. Since the distance function on the hyperboloid and hypersphere only depend on the radius and angles
between points, they are isometric.

We first need to define the projection function $\rho_K$.
For a point $(\xi; {\bm{x}}^T)^T \in \mathbb{R}^{n+1}$ and curvature $K \in \mathbb{R}$, where $\xi \in \mathbb{R}$, ${\bm{x}}, {\bm{y}} \in \mathbb{R}^n$
\begin{align*}\label{eq:rho}
  \rho_K((\xi; {\bm{x}}^T)^T) &= \frac{{\bm{x}}}{1+\sqrt{|K|}\xi},
  &
  \rho_K^{-1}({\bm{y}})&=\left(\frac{1}{\sqrt{|K|}}\frac{1 - K\norm{{\bm{y}}}_2^2}{1 + K\norm{{\bm{y}}}_2^2}; \frac{2{\bm{y}}^T}{1 + K\norm{{\bm{y}}}_2^2}\right)^T.
  \end{align*}
The formulas correspond to the classical stereographic projections defined for these models \citep{curv_intro}. Note that both of these projections map the point
${\bm{\mu}}_0 = (1/\sqrt{|K|}, 0, \ldots, 0)$ in the original space to ${\bm{\mu}}_{0} = \B{0}$ in the projected space, and back.

Since the stereographic projection is conformal, the metric tensors of both spaces will be conformal.
In this case, the metric tensors of both spaces are the same, except for the sign of $K$:
$\mt^{\spr_K}_{\bm{x}} = \mt^{\poi_K}_{\bm{x}} = (\lambda_{\bm{x}}^K)^2 \mt^\euc,$
for all ${\bm{x}}$ in the respective manifold \citep{hnn}, and $\mt_{\bm{y}}^\euc = \eye$ for all ${\bm{y}} \in \euc$.
The conformal factor $\lambda_{\bm{x}}^K$ is then defined as $\lambda_{\bm{x}}^K = 2 / (1 + K\norm{{\bm{x}}}^2_2)$.
Among other things, this form of the metric tensor has the consequence
that we unfortunately cannot define a single unified inner product
in all tangent spaces at all points.
The inner product at ${\bm{x}} \in \mc{M}$ has the form of
$\xprod{{\bm{u}}, {\bm{v}}}{{\bm{x}}} = (\lambda_{\bm{x}}^K)^2 \scprod{{\bm{u}}, {\bm{v}}}$
for all ${\bm{u}}, {\bm{v}} \in \tang{{\bm{x}}}\mc{M}$.

We can now define the two models corresponding to $K > 0$ and $K < 0$.
The curvature of the projected manifold is the same as the original manifold.
An $n$-dimensional \textit{projected hypersphere} ($K > 0$) is defined as the set
$\spr^n_K = \rho_K(\sph_K^n \setminus \{-{\bm{\mu}}_0\}) = \mathbb{R}^n,$
where ${\bm{\mu}}_0 = (1/\sqrt{|K|}, 0, \ldots, 0)^T \in \sph_K^n$, along with the induced distance function.
The $n$-dimensional \textit{Poincar\'{e} ball} $\poi_K^n$ (also called the Poincar\'{e} disk when $n=2$) for a given curvature $K < 0$ is defined as
$\poi^n_K = \rho_K(\hyp_K^n) = \left\{{\bm{x}} \in \mathbb{R}^n : \scprod{{\bm{x}}, {\bm{x}}} < -\frac{1}{K}\right\},$
with the induced distance function.

\subsection{Gyrovector spaces}

An important analogy to vector spaces (vector addition and scalar multiplication) in non-Euclidean geometry is the notion of gyrovector spaces \citep{ungar}.
Both of the above spaces $\spr_K$ and $\poi_K$ (jointly denoted as $\mc{M}_K$) share the same structure, hence they also share the following definition of addition.
The \textit{M\"{o}bius addition} $\kplus$ of ${\bm{x}}, {\bm{y}} \in \mc{M}_K$ (for both signs of $K$) is defined as
\begin{align*}
{\bm{x}} \kplus {\bm{y}} &= \frac{(1-2K\scprod{{\bm{x}}, {\bm{y}}} - K\norm{{\bm{y}}}_2^2){\bm{x}} + (1+K\norm{{\bm{x}}}_2^2){\bm{y}}}{1-2K\scprod{{\bm{x}}, {\bm{y}}} + K^2 \norm{{\bm{x}}}_2^2 \norm{{\bm{y}}}_2^2}.
\end{align*}
We can, therefore, define ``gyrospace distances'' for both of the above spaces, which
are alternative curvature-aware distance functions
\begin{align*}
d_{\spr_\text{gyr}}({\bm{x}}, {\bm{y}}) &= \frac{2}{\sqrt{K}} \arctan(\sqrt{K} \norm{-{\bm{x}} \kplus {\bm{y}}}_2), &
d_{\poi_\text{gyr}}({\bm{x}}, {\bm{y}}) &= \frac{2}{\sqrt{-K}} \arctanh(\sqrt{-K} \norm{-{\bm{x}} \kplus {\bm{y}}}_2).
\end{align*}
These two distances are equivalent to their non-gyrospace variants
$d_{\mc{M}}({\bm{x}}, {\bm{y}}) = d_{\mc{M}_\text{gyr}}({\bm{x}}, {\bm{y}}),$
as is shown in Theorem~\ref{thm:gyro_dist_eq_poi_dist} and its hypersphere equivalent.
Additionally, Theorem~\ref{thm:gyr_dist_poi_converges} shows that
\begin{align*}d_{\mc{M}_\text{gyr}}({\bm{x}}, {\bm{y}}) \xrightarrow{K \to 0} 2 \norm{{\bm{x}}-{\bm{y}}}_2,\end{align*}
which means that the distance functions converge to the Euclidean distance function as $K\to 0$.

We can notice that most statements and operations in constant curvature spaces have a dual statement or operation
in the corresponding space with the opposite curvature sign.
The notion of duality is one which comes up very often and in our case is based on
Euler's formula
$e^{ix} = \cos(x) + i \sin(x)$ and the notion of principal square roots $\sqrt{-K} = i\sqrt{K}$.
This provides a connection between trigonometric, hyperbolic trigonometric, and exponential functions.
Thus, we can convert all the hyperbolic formulas above to their spherical equivalents and vice-versa.

Since \citet{hnn} and \citet{hypglove} used the same gyrovector spaces to define
an exponential map, its inverse logarithmic map, and parallel transport
in the Poincar\'{e} ball, we can reuse them for the projected hypersphere by applying the transformations above, as they share the same formalism.
For parallel transport, we additionally need the notion of gyration \citep{ungar}
\begin{align*}\gyr{{\bm{x}}, {\bm{y}}}{\bm{v}} = \kminus ({\bm{x}} \kplus {\bm{y}}) \kplus ({\bm{x}} \kplus ({\bm{y}} \kplus {\bm{v}})).\end{align*}
Parallel transport in the both the projected hypersphere and the Poincar\'{e} ball then is
$\PT^K_{{\bm{x}} \to {\bm{y}}}({\bm{v}}) = (\lambda_{{\bm{x}}}^K / \lambda_{\bm{y}}^K) \gyr{{\bm{y}}, -{\bm{x}}}{\bm{v}},$
for all ${\bm{x}}, {\bm{y}} \in \mc{M}_K^n$ and ${\bm{v}} \in \tang{{\bm{x}}}\mc{M}_K^n$.
Using a curvature-aware definition scalar products
$\left(\xprod{{\bm{x}}, {\bm{y}}}{K} = \scprod{{\bm{x}}, {\bm{y}}} \text{ if } K \geq 0, \lprod{{\bm{x}}, {\bm{y}}} \text{ if } K < 0\right)$
and of trigonometric functions
\begin{align*}
\sin_K &= \begin{cases}\sin & \text{if } K > 0\\\sinh & \text{if } K < 0\end{cases}& \cos_K &= \begin{cases}\cos & \text{if } K > 0\\\cosh & \text{if } K < 0\end{cases} &
\tan_K &= \begin{cases}\tan & \text{if } K > 0\\\tanh & \text{if } K < 0\end{cases}
\end{align*}
we can summarize all the necessary operations in all manifolds compactly in Table~\ref{tab:summary_ops} and Table~\ref{tab:summary_ops_proj}.

\begin{table}[bt]
\centering
\caption{Summary of operations in $\sph_K$ and $\hyp_K$.}
\label{tab:summary_ops}
\begin{tabular}{ll}
\toprule
Distance & $\displaystyle d({\bm{x}}, {\bm{y}}) = \frac{1}{\sqrt{|K|}} \arccos_K(|K|\xprod{{\bm{x}}, {\bm{y}}}{K})$ \\
\midrule
Exponential map & $\displaystyle \exp_{\bm{x}}^K({\bm{v}}) = \cos_K\left(\sqrt{|K|}\norm{{\bm{v}}}_K\right){\bm{x}} + \sin_K\left(\sqrt{|K|}\norm{{\bm{v}}}_K\right)\frac{{\bm{v}}}{\sqrt{|K|}\norm{{\bm{v}}}_K}$ \\
Logarithmic map & $\displaystyle \log_{\bm{x}}^K({\bm{y}}) = \frac{\arccos_K\left(K \xprod{{\bm{x}}, {\bm{y}}}{K}\right)}{\sin_K\left(\arccos_K\left(K \xprod{{\bm{x}}, {\bm{y}}}{K}\right)\right)} \left({\bm{y}} - K \xprod{{\bm{x}}, {\bm{y}}}{K}{\bm{x}}\right)$\\
\midrule
Parallel transport & $\displaystyle \PT^K_{{\bm{x}} \to {\bm{y}}}({\bm{v}}) = {\bm{v}} - \frac{K\xprod{{\bm{y}}, {\bm{v}}}{K}}{1+K\xprod{{\bm{x}}, {\bm{y}}}{K}}({\bm{x}} + {\bm{y}})$\\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[bt]
\centering
\caption{Summary of operations in projected spaces $\spr_K$ and $\poi_K$.}
\label{tab:summary_ops_proj}
\begin{tabular}{ll}
\toprule
Distance & $\displaystyle d({\bm{x}}, {\bm{y}}) = \frac{1}{\sqrt{|K|}} \arccos_K\left(1 - \frac{2K\norm{{\bm{x}}-{\bm{y}}}_2^2}{(1+K\norm{{\bm{x}}}_2^2)(1+K\norm{{\bm{y}}}_2^2)}\right)$\\
Gyrospace distance & $\displaystyle d_\text{gyr}({\bm{x}}, {\bm{y}}) = \frac{2}{\sqrt{|K|}} \arctan_K(\sqrt{|K|} \norm{-{\bm{x}} \kplus {\bm{y}}}_2)$\\
\midrule
Exponential map & $\displaystyle \exp_{\bm{x}}^K({\bm{v}}) = {\bm{x}} \kplus \left(\tan_K\left(\sqrt{|K|}\frac{\lambda_{\bm{x}}^K\norm{{\bm{v}}}_2}{2}\right)\frac{{\bm{v}}}{\sqrt{|K|}\norm{{\bm{v}}}_2}\right)$ \\
Logarithmic map & $\displaystyle \log_{\bm{x}}^K({\bm{y}}) = \frac{2}{\sqrt{|K|}\lambda_{\bm{x}}^K} \arctan_K\left(\sqrt{|K|}\norm{-{\bm{x}} \kplus {\bm{y}}}_2\right)\frac{-{\bm{x}} \kplus {\bm{y}}}{\norm{-{\bm{x}} \kplus {\bm{y}}}_2}$\\
\midrule
Parallel transport & $\displaystyle \PT^K_{{\bm{x}} \to {\bm{y}}}({\bm{v}}) = \frac{\lambda_{{\bm{x}}}^K}{\lambda_{\bm{y}}^K}\gyr{{\bm{y}}, -{\bm{x}}}{\bm{v}}$\\

\bottomrule
\end{tabular}
\end{table}

\subsection{Products of spaces}\label{bgr:products}

Previously, our space consisted of only one manifold of varying dimensionality
and fixed curvature. Like \citet{products}, we propose learning latent representations in products of constant curvature spaces, contrary to existing VAE approaches which are limited to a single Riemannian manifold.

Our latent space $\mc{M}'$ consists of several \emph{component spaces}
$\mc{M}' = \bigtimes_{i=1}^k \mc{M}_{K_i}^{n_i},$
where $n_i$ is the dimensionality of the space, $K_i$ is its curvature, and $\mc{M} \in \{\euc, \sph, \spr, \hyp, \poi\}$ is the model choice.
Even though all components have constant curvature, the resulting manifold $\mc{M}'$ has non-constant curvature. Its distance function decomposes based on its definition
$d_\mc{M}^2({\bm{x}}, {\bm{y}}) = \sum_{i=1}^k d_{\mc{M}_{K_i}^{n_i}}^2\left({\bm{x}}^{(i)}, {\bm{y}}^{(i)}\right),$
where ${\bm{x}}^{(i)}$ represents a vector in $\mc{M}^{n_i}_{K_i}$, corresponding to the part of the latent space representation of ${\bm{x}}$ belonging to $\mc{M}_{K_i}^{n_i}$.
All other operations we defined on our manifolds are element-wise. Therefore, we again decompose the representations into parts ${\bm{x}}^{(i)}$,
apply the operation on that part
$\tilde{{\bm{x}}}^{(i)} = f_{K_i}^{(n_i)}({\bm{x}}^{(i)})$
and concatenate the resulting parts back $\tilde{{\bm{x}}} = \bigodot_{i=1}^k \tilde{{\bm{x}}}^{(i)}$.

The \emph{signature} of the product space, i.e.~its parametrization, has several degrees of freedom per component:
(i) the model $\mc{M}$, (ii) the dimensionality $n_i$, and (iii) the curvature $K_i$.
We need to select all of the above for every component in our product space.
To simplify, we use a shorthand notation for repeated components: $(\mc{M}_{K_i}^{n_i})^j = \bigtimes_{l=1}^j \mc{M}_{K_i}^{n_i}.$
In Euclidean spaces, the notation is redundant.
For $n_1, \ldots, n_k \in \mathbb{Z}$, such that $\sum_{i=1}^k n_i = n \in \mathbb{Z}$, it holds that the
Cartesian product of Euclidean spaces $\euc^{n_i}$ is
$\euc^n = \bigtimes_{i=1}^k \euc^{n_i}.$
However, the equality does \emph{not} hold for the other considered manifolds. This is due to the additional constraints posed on the points in the definitions of individual models of curved spaces.

\subsection{Probability distributions on Riemannian manifolds}

To be able to train Variational Autoencoders, we
need to chose a probability distribution $p$ as a prior and
a corresponding posterior distribution family $q$.
Both of these distributions have to be differentiable with respect
to their parametrization, they need to have a differentiable
Kullback-Leiber (KL) divergence,
and be ``reparametrizable'' \citep{vae}.
For distributions where the KL does not have a closed-form solution independent on ${\bm{z}}$, or where this integral is too hard to compute,
we can estimate it using Monte Carlo estimation
\begin{align*}D_{\mathrm{KL}}{q}{p} \approx \frac{1}{L} \sum_{l=1}^L \log\left(\frac{q({\bm{z}}^{(l)})}{p({\bm{z}}^{(l)})}\right) \stackrel{\text{if } L=1}{=} \log\left(\frac{q({\bm{z}}^{(1)})}{p({\bm{z}}^{(1)})}\right),\end{align*}
where ${\bm{z}}^{(l)} \sim q$ for all $l = 1, \ldots, L$.
The Euclidean VAE uses a natural choice for a prior
on its latent representations -- the Gaussian distribution \citep{vae}.
Apart from satisfying the requirements for a VAE prior and posterior distribution,
the Gaussian distribution has additional properties, like being the maximum entropy distribution for a given variance \citep{jaynes}.
There exist several fundamentally different approaches to generalizing the Normal distribution to Riemannian manifolds. We discuss the following three generalizations
based on the way they are constructed \citep{hvae}.

\paragraph{Wrapping} This approach leverages the fact that all manifolds define a tangent vector space at every point.
We simply sample from a Gaussian distribution in the tangent space at ${\bm{\mu}}_0$ with mean $\B{0}$, and use parallel transport and the exponential map to map the sampled point onto the manifold. The PDF can be obtained using the multivariate chain rule if we can compute the determinant of the Jacobian of the parallel transport and the exponential map. This is very computationally effective at the expense of losing some theoretical properties.

\paragraph{Restriction} The ``Restricted Normal'' approach is conceptually antagonal -- instead of expanding a point
to a dimensionally larger point, we restrict a point of the ambient space sampled from a Gaussian to the manifold.
The consequence is that the distributions constructed this way
are based on the ``flat'' Euclidean distance.
An example of this is the von Mises-Fisher (vMF) distribution \citep{svae}.
A downside of this approach is that vMF only has a single scalar covariance parameter $\kappa$, while other approaches can parametrize covariance in different dimensions separately.

\paragraph{Maximizing entropy} Assuming a known mean and covariance matrix, we want to maximize the entropy of the distribution \citep{pennec}. This approach is usually called the Riemannian Normal distribution. \citet{hvae} derive it for the Poincar\'{e} ball,
and \citet{hauberg} derive the Spherical Normal distribution on the hypersphere.
Maximum entropy distributions resemble the Gaussian distribution's properties the closest, but it is usually very hard to sample from such distributions, compute their normalization constants, and even
derive the specific form. Since the gains for VAE performance using this construction of Normal distributions over wrapping is only marginal, as reported by \citet{hvae}, we have chosen to focus on Wrapped Normal distributions.

To summarize, Wrapped Normal distributions are very computationally efficient to sample from and also efficient for computing the log probability of a sample, as detailed by \citet{phg}. The Riemannian Normal distributions (based on geodesic distance in the manifold directly) could also be used, however they are more computationally expensive for sampling, because the only methods available are based on rejection sampling \citep{hvae}.

\subsubsection{Wrapped Normal distribution}

First of all, we need to define an ``origin'' point on the manifold, which we will denote as ${\bm{\mu}}_0 \in \mc{M}_K$. What this point corresponds to
is manifold-specific: in the hyperboloid and hypersphere it corresponds to the point
${\bm{\mu}}_0 = (1/\sqrt{|K|}, 0, \ldots, 0)^T,$
and in the Poincar\'{e} ball, projected sphere, and Euclidean space it is simply ${\bm{\mu}}_0 = \B{0},$
the origin of the coordinate system.

Sampling from the distribution $\WN({\bm{\mu}}, {\bm{\Sigma}})$ has been described in detail in \citet{phg} and \citet{hvae}, and we have extended all the necessary operations and procedures to arbitrary curvature $K$. Sampling then corresponds to:
$${\bm{v}} \sim \mc{N}({\bm{\mu}}_0, {\bm{\Sigma}}) \in \tang{{\bm{\mu}}_0}\mc{M}_K,\;{\bm{u}} = \PT^K_{{\bm{\mu}}_0 \to {\bm{\mu}}}({\bm{v}}) \in \tang{{\bm{\mu}}}\mc{M}_K,\;{\bm{z}} = \exp{{\bm{\mu}}}_{\bm{x}}^K({\bm{u}}) \in \mc{M}_K.$$
The log-probability of samples can be computed by the reverse procedure:
\begin{align*}
{\bm{u}} = \log_\mu^K({\bm{z}}) \in \tang{{\bm{\mu}}}\mc{M}_K, \; {\bm{v}} = \PT^K_{{\bm{\mu}} \to {\bm{\mu}}_0}({\bm{u}}) \in \tang{{\bm{\mu}}_0}\mc{M}_K,\\
\log \WN({\bm{\mu}}, {\bm{\Sigma}}) = \log\mc{N}({\bm{v}}; {\bm{\mu}}_0, {\bm{\Sigma}}) - \log\det\left(\frac{\partial f}{\partial {\bm{v}}}\right),
\end{align*}
where $f = \exp_{\bm{\mu}}^K \circ \PT^K_{{\bm{\mu}}_0 \to {\bm{\mu}}}.$
The distribution can be applied to all manifolds that we have introduced. The only differences are the specific forms of operations and the log-determinant in the PDF.
The specific forms of the log-PDF for the four spaces $\hyp$, $\sph$, $\spr$, and $\poi$ are derived in Appendix~\ref{app:prob}.
All variants of this distribution are reparametrizable,
differentiable, and the KL can be computed using Monte Carlo estimation.
As a consequence of the distance function and operations convergence theorems for the Poincar\'e ball \ref{thm:gyr_dist_poi_converges} (analogously for the projected hypersphere),
\ref{thm:exp_conv}, \ref{thm:log_conv}, and \ref{thm:pt_conv},
we see that the Wrapped Normal distribution converges
to the Gaussian distribution as $K \to 0$.

\section{Variational Autoencoders in products spaces}\label{chap:vae}

To be able to learn latent representations in Riemannian manifolds instead of
in $\mathbb{R}^d$ as above, we only need to change the parametrization of the mean and covariance in the VAE forward pass,
and the choice of prior and posterior distributions.
The prior and posterior have to be chosen depending on the chosen manifold
and are essentially treated as hyperparameters of our VAE.
Since we have defined the Wrapped Normal family of distributions
for all spaces, we can use $\WN({\bm{\mu}}_0, {\bm{\sigma}}^2\eye)$ as the posterior family,
and $\WN({\bm{\mu}}_0, \eye)$ as the prior distribution. The forms of the distributions
depend on the chosen space type.
The mean is parametrized using the exponential map $\exp^K_{{\bm{\mu}}_0}$ as an activation function.
Hence, all the model's parameters live in Euclidean space and can be optimized directly.

In experiments, we sometimes use $\vmf({\bm{\mu}}, \kappa)$ for the hypersphere $\sph_K^n$ (or a backprojected variant of $\vmf$ for $\spr_K^n$) with the associated hyperspherical uniform distribution $U(\sph_K^n)$ as a prior \citep{svae},
or the Riemannian Normal distribution $\RN({\bm{\mu}}, \sigma^2)$ and the associated prior $\RN{{\bm{\mu}}_0, 1}$ for the Poincare ball $\poi_K^n$ \citep{hvae}.

\subsection{Learning curvature}\label{chap:curvature}

We have already seen approaches to learning VAEs in products of spaces of constant curvature.
However, we can also change the curvature constant in each of the spaces during training.
The individual spaces will still have constant curvature at each point, we just allow changing the constant in between
training steps. To differentiate between these training procedures,
we will call them \emph{fixed} curvature and \emph{learnable} curvature VAEs respectively.

The motivation behind changing curvature of non-Euclidean constant curvature spaces might not be clear,
since it is apparent from the definition of the distance function in the hypersphere and hyperboloid
$d({\bm{x}}, {\bm{y}}) = R \cdot \theta_{{\bm{x}}, {\bm{y}}},$
that the distances between two points that stay at the same angle only get rescaled when changing the radius of the space. Same applies for the Poincar\'{e} ball and the projected spherical space.
However, the decoder does not only depend on pairwise distances, but rather on the specific positions of points in the space. It can be
conjectured that the KL term of the ELBO indeed is only ``rescaled'' when we change the curvature,
however, the reconstruction process is influenced in non-trivial ways. Since that is hard to quantify and prove, we devise a series of practical experiments to show overall model performance is enhanced when learning curvature.

\paragraph{Fixed curvature VAEs}

In \emph{fixed} curvature VAEs, all component latent spaces have a fixed curvature
that is selected a priori and fixed for the whole duration of the training procedure, as well as during evaluation.
For Euclidean components it is 0, for positively or negatively curved spaces any positive
or negative number can be chosen, respectively. For stability reasons, we select curvature values
from the range $[0.25, 1.0]$, which corresponds to radii in $[1.0, 2.0]$.
The exact curvature value does not have a significant impact on performance when training a fixed curvature VAE, as motivated by the distance rescaling remark above. In the following, we refer to fixed curvature components with a constant subscript, e.g.~$\hyp_1^n$.

\paragraph{Learnable curvature VAEs}

In all our manifolds, we can differentiate the ELBO
with respect to the curvature $K$. This enables us to treat $K$ as a parameter of the model
and learn it using gradient-based optimization, exactly like we learn the encoder/decoder
maps in a VAE.

Learning curvature directly is badly conditioned --
we are trying to learn one scalar parameter that influences the resulting decoder
and hence the ELBO quite heavily.
Empirically, we have found that Stochastic Gradient Descent works well to optimize the radius of a component. We constrain the radius to be strictly positive in all non-Euclidean spaces
by applying a ReLU activation function before we use it in operations.

\paragraph{Universal curvature VAEs}

However, we must still a priori select the ``partitioning'' of our latent space -- the number of components and for each of them select the dimension and at least the sign of the curvature of that component (signature estimation).

The simplest approach would be to just try all possibilities and compare the results
on a specific dataset. This procedure would most likely be optimal, but does not scale well.

To eliminate this, we propose an approximate method --
we partition our space into 2-dimensional components (if the number of dimensions is odd, one component will have 3 dimensions).
We initialize all of them as Euclidean components and train for
half the number of maximal epochs we are allowed.
Then, we split the components into 3 approximately equal-sized groups
and make one group into hyperbolic components, one into spherical, and the last
remains Euclidean. We do this by changing the curvature of a component by a very small ${\epsilon}$.
We then train just the encoder/decoder maps for a few epochs to stabilize
the representations after changing the curvatures. Finally, we allow learning the curvatures of all non-Euclidean components and train for the rest of the allowed epochs.
The method is not completely general, as it never uses components bigger than dimension 2, but the approximation has empirically performed satisfactorily.

We also do not constrain the curvature of the components to a specific sign in the last stage of training.
Therefore, components may change their type of space from a positively curved to a negatively curved one, or vice-versa.

Because of the divergence of points as $K \to 0$ for the hyperboloid and hypersphere, the universal curvature VAE assumes the positively curved
space is $\spr$ and the negatively curved space is $\poi$. In all experiments, this universal approach is denoted as $\uni^n$.

\section{Experiments}\label{chap:exp}

For our experiments, we use four datasets:
(i) Branching diffusion process \citep[BDP]{hvae} -- a synthetic tree-like dataset with injected noise,
(ii) Dynamically-binarized MNIST digits \citep{mnist} -- we binarize the images similarly to \citet{burda,dbns}:
  the training set is binarized dynamically (uniformly sampled threshold per-sample $\mathrm{bin}({\bm{x}}) \in \{0,1\}^D = {\bm{x}} > \mc{U}[0,1], {\bm{x}} \in {\bm{x}} \subseteq [0,1]^D$),
  and the evaluation set is done with a fixed binarization (${\bm{x}} > 0.5$),
(iii) Dynamically-binarized Omniglot characters \citep{omniglot} downsampled to $28 \times 28$ pixels, and
(iv) CIFAR-10 \citep{cifar}.

All models in all datasets are trained with early stopping on training ELBO with a lookahead of 50 epochs and a warmup of 100 epochs \citep{nvrnn}. All BDP models are trained for a 1000 epochs, MNIST and Omniglot models are trained for 300 epochs,
and CIFAR for 200 epochs.
We compare models with a given latent space dimension using marginal log-likelihood with importance sampling \citep{burda}
with 500 samples, except for CIFAR, which uses 50 due to memory constraints.
In all tables, we denote it as LL. We run all experiments at least 3 times to get an estimate of variance when using different
initial values.

In all the BDP, MNIST, and Omniglot experiments below, we use a simple feed-forward encoder and decoder architecture consisting of a single dense layer with 400 neurons and element-wise ReLU activation.
Since all the VAE parameters $\{\theta, \phi\}$ live in Euclidean manifolds, we can use standard gradient-based optimization methods.
Specifically, we use the Adam \citep{adam} optimizer with a learning rate of $10^{-3}$ and standard settings for $\beta_1 = 0.9$, $\beta_2 = 0.999$, and $\epsilon = 10^{-8}$.

For the CIFAR encoder map, we use a simple convolutional neural networks with three convolutional
layers with 64, 128, and 512 channels respectively.
For the decoder map, we first use a dense layer of dimension 2048,
and then three consecutive transposed convolutional layers with
256, 64, and 3 channels.
All layers are followed by a ReLU activation function, except for the last one.
All convolutions have $4 \times 4$ kernels with stride 2, and padding of size 1.

The first 10 epochs for all models are trained with a fixed curvature starting at 0
and increasing in absolute value each epoch. This corresponds to a burn-in period similarly to \citet{hypemb}.
For learnable curvature approaches we then use Stochastic Gradient Descent with learning rate $10^{-4}$ and let the optimizers adjust the value freely, for fixed curvature approaches it stays at the last burn-in value.
All our models use the Wrapped Normal distribution, or equivalently Gaussian in Euclidean components, unless specified otherwise.
All fixed curvature components are denoted with a $\mc{M}_1$ or $\mc{M}_{-1}$ subscript,
learnable curvature components do not have a subscript.
The observation model for the reconstruction loss term were Bernoulli distributions for MNIST and Omniglot, and standard Gaussian distributions for BDP and CIFAR.

As baselines, we train VAEs with spaces that have a fixed constant curvature,
i.e.~assume a single Riemannian manifold (potentially a product of them) as their latent space.
It is apparent that our models with a single component, like $\sph_1^n$ correspond to \citet{svae} and \citet{svae2}, $\hyp_{-1}^n$ is equivalent to the Hyperbolic VAE of \citet{phg},
$\poi_{-c}^n$ corresponds to the $\mc{P}^c$-VAE of \citet{hvae},
and $\euc^n$ is equivalent to the Euclidean VAE.
In the following, we present a selection of all the obtained results. For more information see Appendix~\ref{app:res}. Bold numbers represent values that are particularly
interesting.
Since the Riemannian Normal and the von Mises-Fischer distribution only have a spherical covariance matrix, i.e.~a single scalar variance parameter per component, we evaluate all our approaches with a spherical covariance parametrization as well.

\begin{table}[tbp]
  \centering
  \caption{Summary of results (mean and standard deviation), latent space dimension 6, spherical covariance, on the BDP (left) and MNIST (right) datasets.}
  \label{tab:bdp_z6_scalar_main}
  \begin{tabular}[t]{l|r}
    \toprule
    \textbf{Model} & LL\\
    \midrule
$\sph_{1}^{6}$ & $-55.81${\scriptsize$\pm 0.35$}\\
$\spr_{1}^{6}$ & $-55.78${\scriptsize$\pm 0.07$}\\
$\euc^{6}$ & $-56.28${\scriptsize$\pm 0.56$}\\
$(\hyp_{-1}^{2})^{3}$ & $-56.08${\scriptsize$\pm 0.52$}\\
$\hyp_{-1}^{6}$ & $-56.18${\scriptsize$\pm 0.32$}\\
$(\poi_{-1}^{2})^{3}$ & $-55.98${\scriptsize$\pm 0.62$}\\
$\poi_{-1}^{6}$ & $-56.74${\scriptsize$\pm 0.55$}\\
$(\RN\;\poi_{-1}^{2})^{3}$ & $\mathbf{-54.99}${\scriptsize$\pm 0.12$}\\
\midrule
$(\sph^{2})^{3}$ & $-56.05${\scriptsize$\pm 0.21$}\\
$(\spr^{2})^{3}$ & $-56.06${\scriptsize$\pm 0.36$}\\
$(\hyp^{2})^{3}$ & $\mathbf{-55.80}${\scriptsize$\pm 0.32$}\\
$(\poi^{2})^{3}$ & $-56.29${\scriptsize$\pm 0.05$}\\
$(\RN\;\poi^{2})^{3}$ & $-56.25${\scriptsize$\pm 0.56$}\\
\midrule
$\spr^{2} \times \euc^{2} \times \poi^{2}$ & $-55.87${\scriptsize$\pm 0.22$}\\
$\euc^{2} \times \hyp^{2} \times \sph^{2}$ & $-55.92${\scriptsize$\pm 0.42$}\\
$\euc^{2} \times \hyp^{2} \times (\vmf\;\sph^{2})$ & $-55.82${\scriptsize$\pm 0.43$}\\
$\euc^{2} \times \hyp_{-1}^{2} \times (\vmf\;\sph_{1}^{2})$ & $\mathbf{-55.77}${\scriptsize$\pm 0.51$}\\
\midrule
$(\uni^{2})^{3}$ & $\mathbf{-55.56}${\scriptsize$\pm 0.15$}\\
$\uni^{6}$ & $-55.84${\scriptsize$\pm 0.38$}\\
    \bottomrule
  \end{tabular}
  ~
  \begin{tabular}[t]{l|r}
    \toprule
    \textbf{Model} & LL\\
\midrule
$\sph_{1}^{6}$ & $-96.71${\scriptsize$\pm 0.17$}\\
$\vmf\;\sph_{1}^{6}$ & $-97.03${\scriptsize$\pm 0.14$}\\
$\spr_{1}^{6}$ & $-98.21${\scriptsize$\pm 0.23$}\\
$\euc^{6}$ & $-97.16${\scriptsize$\pm 0.15$}\\
$\hyp_{-1}^{6}$ & $-97.10${\scriptsize$\pm 0.44$}\\
$(\poi_{-1}^{2})^{3}$ & $-97.56${\scriptsize$\pm 0.04$}\\
$(\RN\;\poi_{-1}^{2})^{3}$ & $\mathbf{-92.54}${\scriptsize$\pm 0.19$}\\
\midrule
$(\sph^{2})^{3}$ & $-96.46${\scriptsize$\pm 0.12$}\\
$\sph^{6}$ & $-96.72${\scriptsize$\pm 0.15$}\\
$\vmf\;\sph^{6}$ & $-96.72${\scriptsize$\pm 0.18$}\\
$\spr^{6}$ & $-97.72${\scriptsize$\pm 0.15$}\\
$(\hyp^{2})^{3}$ & $-97.37${\scriptsize$\pm 0.13$}\\
$(\RN\;\poi^{2})^{3}$ & $\mathbf{-94.16}${\scriptsize$\pm 0.68$}\\
\midrule
$\spr^{2} \times \euc^{2} \times \poi^{2}$ & $-97.48${\scriptsize$\pm 0.18$}\\
$\spr^{2} \times \euc^{2} \times (\RN\;\poi^{2})$ & $-96.43${\scriptsize$\pm 0.47$}\\
$\spr_{1}^{2} \times \euc^{2} \times (\RN\;\poi^{2}_{-1})$ & $\mathbf{-96.18}${\scriptsize$\pm 0.21$}\\
$\euc^{2} \times \hyp^{2} \times \sph^{2}$ & $-96.80${\scriptsize$\pm 0.20$}\\
$\euc^{2} \times \hyp_{-1}^{2} \times \sph_{1}^{2}$ & $-96.76${\scriptsize$\pm 0.09$}\\
$\euc^{2} \times \hyp^{2} \times (\vmf\;\sph^{2})$ & $-96.56${\scriptsize$\pm 0.27$}\\
\midrule
$(\uni^{2})^{3}$ & $\mathbf{-97.12}${\scriptsize$\pm 0.04$}\\
$\uni^{6}$ & $-97.26${\scriptsize$\pm 0.16$}\\
    \bottomrule
  \end{tabular}
\end{table}

\paragraph{Binary diffusion process}
For the BDP dataset and latent dimension 6 (Table~\ref{tab:bdp_z6_scalar_main}),
we observe that all VAEs that only use the von Mises-Fischer distribution
perform worse than a Wrapped Normal. However, when a VMF spherical
component was paired with other component types, it performed better
than if a Wrapped Normal spherical component was used instead.
Riemannian Normal VAEs did very well on their own -- the fixed Poincar\'{e}
VAE $(\RN\;\poi^2_{-1})^3$ obtains the best score. It did not fare
as well when we tried to learn curvature with it, however.

An interesting observation is that all single-component VAEs $\mc{M}^6$ performed worse than product VAEs $(\mc{M}^2)^3$
when curvature was learned, across all component types.
Our universal curvature VAE $(\uni^2)^3$ managed to get better results
than all other approaches except for the Riemannian Normal baseline,
but it is within the margin of error of some other models.
It also outperformed its single-component variant $\uni^6$.
However, we did not find that it converged to specific curvature values,
only that they were in the approximate range of $(-0.1, +0.1)$.

\paragraph{Dynamically-binarized MNIST reconstruction}
On MNIST (Table~\ref{tab:bdp_z6_scalar_main})
with spherical covariance, we noticed that VMF again under-performed
Wrapped Normal, except when it was part of a product like $\euc^{2} \times \hyp^{2} \times (\vmf\;\sph^{2})$.
When paired with another Euclidean and a Riemannian Normal Poincar\'{e} disk component,
it performed well, but that might be because the $\RN\;\poi_{-1}$
component achieved best results across the board on MNIST. It achieved good
results even compared to diagonal covariance VAEs on 6-dimensional MNIST.
Several approaches are better than the Euclidean baseline.
That applies mainly to the above mentioned Riemannian Normal Poincar\'{e} ball components,
but also $\sph^6$ both with Wrapped Normal and VMF, as well as most
product space VAEs with different curvatures (third section of the table).
Our $(\uni^2)^3$ performed similarly to the Euclidean baseline VAE.

\begin{table}[tb]
  \centering
  \caption{Summary of selected models (mean and standard deviation), latent space dimension 6 (left) and 12 (right), diagonal covariance, on the MNIST dataset.}
  \label{tab:mnist_z6_diag_main}

  \begin{tabular}[t]{l|r}
    \toprule
    \textbf{Model} & LL\\
    \midrule
$\sph_{1}^{6}$ & $\mathbf{-96.51}${\scriptsize$\pm 0.09$}\\
$\spr_{1}^{6}$ & $-97.89${\scriptsize$\pm 0.10$}\\
$\euc^{6}$ & $-96.88${\scriptsize$\pm 0.16$}\\
$\hyp_{-1}^{6}$ & $-97.38${\scriptsize$\pm 0.73$}\\
$\poi_{-1}^{6}$ & $-97.33${\scriptsize$\pm 0.15$}\\
\midrule
$\sph^{6}$ & $\mathbf{-96.44}${\scriptsize$\pm 0.20$}\\
$\spr^{6}$ & $-97.53${\scriptsize$\pm 0.22$}\\
$(\hyp^{2})^{3}$ & $\mathbf{-96.86}${\scriptsize$\pm 0.31$}\\
$\hyp^{6}$ & $-96.90${\scriptsize$\pm 0.26$}\\
$\poi^{6}$ & $-97.26${\scriptsize$\pm 0.16$}\\
\midrule
$\spr^{2} \times \euc^{2} \times \poi^{2}$ & $-97.37${\scriptsize$\pm 0.14$}\\
$\spr_{1}^{2} \times \euc^{2} \times \poi_{-1}^{2}$ & $-97.29${\scriptsize$\pm 0.16$}\\
$\euc^{2} \times \hyp^{2} \times \sph^{2}$ & $\mathbf{-96.71}${\scriptsize$\pm 0.19$}\\
$\euc^{2} \times \hyp_{-1}^{2} \times \sph_{1}^{2}$ & $\mathbf{-96.66}${\scriptsize$\pm 0.27$} \\
\midrule
$(\uni^{2})^{3}$ & $-97.06${\scriptsize$\pm 0.13$}\\
$\uni^{6}$ & $\mathbf{-96.90}${\scriptsize$\pm 0.10$}\\
    \bottomrule
  \end{tabular}
  ~
  \begin{tabular}[t]{l|r}
    \toprule
    \textbf{Model} & LL\\
    \midrule
$(\sph_{1}^{2})^{6}$ & $-79.92${\scriptsize$\pm 0.21$}\\
$(\spr_{1}^{2})^{6}$ & $-80.53${\scriptsize$\pm 0.10$}\\
$\euc^{12}$ & $\mathbf{-79.51}${\scriptsize$\pm 0.09$}\\
$(\hyp_{-1}^{2})^{6}$ & $-80.54${\scriptsize$\pm 0.23$}\\
$\hyp_{-1}^{12}$ & $\textbf{-79.37}${\scriptsize$\pm 0.14$}\\
$(\poi_{-1}^{2})^{6}$ & $-80.39${\scriptsize$\pm 0.07$}\\
\midrule
$\sph^{12}$ & $-79.99${\scriptsize$\pm 0.27$}\\
$\spr^{12}$ & $-80.37${\scriptsize$\pm 0.16$}\\
$\hyp^{12}$ & $-79.77${\scriptsize$\pm 0.10$}\\
$(\poi^{2})^{6}$ & $-80.31${\scriptsize$\pm 0.08$}\\
\midrule
$(\spr_{1}^{2})^{2} \times (\euc^{2})^{2} \times (\poi_{-1}^{2})^{2}$ & $-80.14${\scriptsize$\pm 0.11$}\\
$\spr_{1}^{4} \times \euc^{4} \times \poi_{-1}^{4}$ & $-80.14${\scriptsize$\pm 0.20$}\\
$(\euc^{2})^{2} \times (\hyp^{2})^{2} \times (\sph^{2})^{2}$ & $\mathbf{-79.59}${\scriptsize$\pm 0.25$}\\
$\euc^{4} \times \hyp^{4} \times \sph^{4}$ & $\mathbf{-79.69}${\scriptsize$\pm 0.14$}\\
\midrule
$(\uni^{2})^{6}$ & $\mathbf{-79.61}${\scriptsize$\pm 0.06$}\\
$\uni^{12}$ & $-80.01${\scriptsize$\pm 0.30$}\\
    \bottomrule
  \end{tabular}
\end{table}

With diagonal covariance parametrization (Table~\ref{tab:mnist_z6_diag_main}), we observe similar trends.
With a latent dimension of 6, the Riemannian Normal Poincar\'{e} ball VAE is still the best performer.
The Euclidean baseline VAE achieved better results than its spherical covariance counterpart.
Overall, the best result is achieved by the single-component spherical model, with learnable
curvature $\sph_6$.
Interestingly, all single-component VAEs performed better than their $(\mc{M}^2)^3$ counterparts,
except for the $\hyp^6$ hyperboloid, but only by a tiny margin.
Products of different component types also achieve good results.
Noteworthy is that their fixed curvature variants seem to perform marginally better than learnable
curvature ones. Our universal VAEs perform at around the Euclidean baseline VAE performance.
Interestingly, all of them end up with negative curvatures $-0.3 < K < 0$.

Secondly, we run our models with a latent space dimension of 12.
We immediately notice, that not many models can beat the Euclidean VAE baseline $\euc^{12}$
consistently, but several are within the margin of error.
Notably, the product VAEs of $\hyp$, $\sph$, and $\euc$,
fixed and learnable $\hyp^{12}$, and our universal VAE $(\uni^2)^6$.
Interestingly, products of small components perform better when curvature is fixed, whereas
single big component VAEs are better when curvature is learned.

\paragraph{Dynamically-binarized Omniglot reconstruction}
For a latent space of dimension 6 (Table~\ref{tab:omni_z6_diag_main}), the best of the baseline models is the Poincar\'{e} VAE of
\citep{hvae}. Our models that come very close to the average estimated marginal log-likelihood,
and are definitely within the margin of error, are mainly $(\sph^2)^3$,
$\spr^{2} \times \euc^{2} \times \poi^{2}$, and $\uni^6$.
However, with the variance of performance across different runs,
we cannot draw a clear conclusion. In general, hyperbolic VAEs seem to be doing a bit better on this dataset than spherical VAEs,
which is also confirmed by the fact that almost all universal curvature models finished with
negative curvature components.

\begin{table}[tb]
  \centering
  \caption{Summary of selected models (mean and standard deviation), latent space dimension 6, diagonal covariance, on the Omniglot (left) and CIFAR-10 dataset (right), respectively.}
  \label{tab:omni_z6_diag_main}

  \begin{tabular}[t]{l|r}
    \toprule
    \textbf{Model} & LL\\
\midrule
$\sph_{1}^{6}$ & $-136.69${\scriptsize$\pm 0.94$}\\
$\spr_{1}^{6}$ & $-137.42${\scriptsize$\pm 1.20$}\\
$\euc^{6}$ & $-136.05${\scriptsize$\pm 0.29$}\\
$\hyp_{-1}^{6}$ & $-137.09${\scriptsize$\pm 0.06$}\\
$\poi_{-1}^{6}$ & $\mathbf{-135.86}${\scriptsize$\pm 0.20$}\\
\midrule
$(\sph^{2})^{3}$ & $\mathbf{-136.14}${\scriptsize$\pm 0.27$}\\
$\sph^{6}$ & $-136.20${\scriptsize$\pm 0.44$}\\
$(\spr^{2})^{3}$ & $-136.13${\scriptsize$\pm 0.17$}\\
$\spr^{6}$ & $-136.30${\scriptsize$\pm 0.08$}\\
$(\hyp^{2})^{3}$ & $-136.17${\scriptsize$\pm 0.09$}\\
$\hyp^{6}$ & $-136.24${\scriptsize$\pm 0.32$}\\
$(\poi^{2})^{3}$ & $-136.09${\scriptsize$\pm 0.07$}\\
$\poi^{6}$ & $-136.05${\scriptsize$\pm 0.44$}\\
\midrule
$\spr^{2} \times \euc^{2} \times \poi^{2}$ & $\mathbf{-135.89}${\scriptsize$\pm 0.40$}\\
$\euc^{2} \times \hyp^{2} \times \sph^{2}$ & $-135.93${\scriptsize$\pm 0.48$}\\
\midrule
$(\uni^{2})^{3}$ & $-136.21${\scriptsize$\pm 0.07$}\\
$\uni^{6}$ & $\mathbf{-136.04}${\scriptsize$\pm 0.17$}\\
    \bottomrule
  \end{tabular}
  ~
  \begin{tabular}[t]{l|r}
    \toprule
    \textbf{Model} & LL\\
\midrule
$\euc^{6}$ & $-1896.19${\scriptsize$\pm 2.54$}\\
$\hyp_{-1}^{6}$ & $\mathbf{-1888.23}${\scriptsize$\pm 2.12$}\\
$\poi_{-1}^{6}$ & $-1893.27${\scriptsize$\pm 0.61$}\\
\midrule
$\spr^{6}$ & $-1893.85${\scriptsize$\pm 0.36$}\\
$\sph^{6}$ & $\mathbf{-1889.76}${\scriptsize$\pm 1.62$}\\
$\poi^{6}$ & $-1891.40${\scriptsize$\pm 2.14$}\\
$\spr^{2} \times \euc^{2} \times \poi^{2}$ & $-1899.90${\scriptsize$\pm 4.60$}\\
$\euc^{2} \times \hyp^{2} \times \sph^{2}$ & $-1895.46${\scriptsize$\pm 0.92$}\\
$(\uni^{2})^{3}$ & $-1895.09${\scriptsize$\pm 4.27$}\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{CIFAR-10 reconstruction}

For a latent space of dimension 6, we can observe that
almost all non-Euclidean models perform better than the euclidean baseline $\euc^6$.
Especially well-performing is the fixed hyperboloid $\hyp^6_{-1}$,
and the learnable hypersphere $\sph^6$.
Curvatures for all learnable models on this dataset converge to values in the approximate range of $(-0.15, +0.15)$.

\paragraph{Summary}
In conclusion, a very good model seems to be the Riemannian Normal Poincar\'{e} ball VAE $\RN\;\poi^n$.
However, it has practical limitations due to a rejection sampling algorithm
and an unstable implementation.
On the contrary, von Mises-Fischer spherical VAEs have almost consistently performed
worse than their Wrapped Normal equivalents. Overall, Wrapped Normal VAEs
in all constant curvature manifolds
seem to perform well at modeling the latent space.

A key takeaway is that our universal curvature models $\uni^n$ and $(\uni^2)^{\floor{n/2}}$
seem to generally outperform their corresponding Euclidean VAE baselines
in lower-dimensional latent spaces and, with minor losses,
manage to keep most of the competitive performance as the dimensionality goes up,
contrary to VAEs with other non-Euclidean components.

\section{Conclusion}\label{chap:concl}

By transforming the latent space and associated prior distributions
onto Riemannian manifolds of constant curvature, it has previously been shown that
we can learn representations on curved space.
Generalizing on the above ideas, we have extended the theory of learning
VAEs to products of constant curvature spaces.
To do that, we have derived the necessary operations in several models of constant curvature spaces,
extended existing probability distribution families to these manifolds,
and generalized VAEs to latent spaces that are products of smaller ``component'' spaces, with learnable curvature.
On various datasets, we show that our approach is competitive and additionally has the property that it generalizes the Euclidean variational autoencoder --
if the curvatures of all components go to 0, we recover the VAE of \citet{vae}.

\subsubsection*{Acknowledgments}
We would like to thank Andreas Bloch for help in verifying some of the formulas for constant
curvature spaces and for many insightful discussions; Prof.~Thomas Hofmann and the Data Analytics Lab, the Leonhard cluster, and
ETH Z\"{u}rich for GPU access.

Work was done while all authors were at ETH Z\"urich.

Ondrej Skopek (\href{mailto:oskopek@google.com}{\texttt{oskopek@google.com}}) is now at Google.

Octavian-Eugen Ganea (\href{mailto:oct@mit.edu}{\texttt{oct@mit.edu}}) is now at the Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology.

Gary B\'{e}cigneul (\href{mailto:gary.becigneul@inf.ethz.ch}{\texttt{gary.becigneul@inf.ethz.ch}}) is funded by the Max Planck ETH Center for Learning Systems.

redpajama_set_name: RedPajamaArXiv


text:
Q: Need help adding events to a Javascript Calendar I'm trying to create a simple event calendar that has a changing background for each month. I have looked at fullCalendar, but couldn't figure out how to add a background to each month. So I found this calendar code: http://www.java2s.com/Code/JavaScript/GUI-Components/Calendarwithimageforeachmonth.htm
I'm using the above code, and I've modified it to create a different background for each month. Now all I need to do is add about 8 events to the calendar. I know that fullCalendar does this via the following code:
    $(document).ready(function() {

    var date = new Date();
    var d = date.getDate();
    var m = date.getMonth();
    var y = date.getFullYear();

    $('#calendar').fullCalendar({
        editable: true,
        events: [
            {
                title: 'All Day Event',
                start: new Date(y, m, 1)
            },
            {
                title: 'Long Event',
                start: new Date(y, m, d-5),
                end: new Date(y, m, d-2)
            },
            {
                id: 999,
                title: 'Repeating Event',
                start: new Date(y, m, d-3, 16, 0),
                allDay: false
            },
            {
                id: 999,
                title: 'Repeating Event',
                start: new Date(y, m, d+4, 16, 0),
                allDay: false
            },
            {
                title: 'Meeting',
                start: new Date(y, m, d, 10, 30),
                allDay: false
            },
            {
                title: 'Lunch',
                start: new Date(y, m, d, 12, 0),
                end: new Date(y, m, d, 14, 0),
                allDay: false
            },
            {
                title: 'Birthday Party',
                start: new Date(y, m, d+1, 19, 0),
                end: new Date(y, m, d+1, 22, 30),
                allDay: false
            },
            {
                title: 'Click for Google',
                start: new Date(y, m, 28),
                end: new Date(y, m, 29),
                url: 'http://google.com/'
            }
        ]
    });

});

I am relatively new to editing/learning javascript, so I'm not sure where to begin. I am trying to incorporate that fullCalendar code into the calendar that I've modified from the code in the above link.
For example, I want to add "Independance Day" to July 4th.
Any help is much appreciated!
Thanks!

A: You just need to add an event to a specific date you want to .
like,
 events: [
        {
            title: 'Independence Day',
            start: new Date(2014, 06, 4)
        }];

This will add a event for the independence day to your calendar.

redpajama_set_name: RedPajamaStackExchange


text:
Located in the Linden Hills neighborhood of Minneapolis, less than a mile from 50th & France in Edina, Lake Harriet Law Office, LLC represents clients exclusively in the areas of Divorce and Family Law. The top financial issues during Divorce include Asset Valuation, Property Division, Marital Cash Flow Analysis, Balance Sheet Preparation, Debt Allocation, Real Estate and Business Valuation, Spousal Maintenance, Tax Issues and related Financial Matters. For parties with minor children, the resolution of Parenting Time, Child Custody and Child Support provides a solid framework for successful Co-Parenting post-decree. Please contact us to schedule a consultation.
During a Divorce, it is important to retain a highly experienced law firm that has significant expertise with financial matters including Property Division, the valuation of Assets and Debts, Spousal Maintenance (Alimony), Real Estate issues, Cash Flow Schedules, Balance Sheet Preparation, Business Valuation, Present Value Calculations, the Analysis of Retirement Accounts, and Tax Issues associated with Divorce. In Minnesota, the division of Assets and Debts must be equitable, and not necessarily equal. We are a numbers savvy law firm that focuses on the needs of the client. Please contact us to schedule an office consultation.
For parties with minor children, Divorce includes more than just the division of assets and debts, it also includes Parenting Time, Child Custody and other issues that impact children. It is important to retain a highly experienced Divorce and Family Law firm who can effectively advocate your position during the case, and who understands every phase of the Family Law process. Approximately 95% of cases settle through agreements reached by the parties. Alternative Dispute Resolution, including Mediation and the Early Neutral Evaluation (ENE) process, is part of most contested Divorce and Family Law cases.
Latest Family Law Updates
11.20.2020by Randall A. Smith
Spousal Maintenance, the 8 Factors and Equalizing Income
11.20.2020by Esther Raty
Reconsidering Family Finances in COVID-19
Congratulations to Katie on her new law clerk job with Chief Judge Stoney Hiljus
a consultation

redpajama_set_name: RedPajamaCommonCrawl


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Condo Watch
175 Hunter Condos
175 Hunter Street East, Hamilton
80 units (6 floors)
Downtown Hamilton
Built in 1990 by Centre Gate Properties Ltd, 175 Hunter Street East is a 6 floor building offering 80 units. Floor plans range from 550 to 913 square feet and offer in suite laundry and a no-smoking building environment.
Residents have access to a rooftop patio area with city views and party room.
175 Hunter Condos is located in trendy Corktown providing walking distance access to trendy restaurants night life, grocery options (Denninger's and Nations Fresh Foods). Steps to public transit routes that connect you to the rest of the city along with the Hunter GO Station.
Floorplan Options
(550 - 913 sq ft)
1 Bedroom, 2 Bedroom
Party Room, Rooftop Terrace
($313 - $520 per month)
Common Element Maintenance and Insurance, Water, Parking
Pricing: $190,000 to $315,000
Search Hamilton Condos For Sale or Rent
Other Condo Options In Downtown Hamilton
The Durand - 120 Duke Street
The Durand is a 15 floor condominium located at 120 Duke Street in Hamilton. It has 113 units ranging from 935 to 1,275 square feet.
The Crofts - 141 Catharine Street South
The Crofts is a condominium conversion located at 141 Catharine St S in Downtown Hamilton. Located in trendy Corktown it offers affordable one bedroom units.
Queensvilla Condominiums - 101 Queen Street South
Queensvilla Condominiums is an eight floor condo conversion at 101 Queen Street South in downtown Hamilton. It offers 47 one bedroom units.
The Residences of Royal Connaught - 112 King Street East
Completed in 1916, the Royal Connaught has been reborn as a luxury condominium development in the heart of Hamilton. Modern floor plans and over 35,000 square feet of amenity space.
City Square - 90 Charlton Avenue West
Situated in the Durnand District of downtown Hamilton, City Square Condos at 90 Charlton Avenue West houses 76 units in the 9-storey first tower. It was the first new-build condo development in the downtown area in over 20 years.
The Pigott Building - 36 James Street South
Built in 1928 and developed into residential condominiums in 1996, this historical 18-storey building was Hamilton's first skyscraper. Located in the heart of downtown Hamilton it offers one and two storey floor plans ranging between one and three bedrooms.
101 Locke Condos - 101 Locke Street South
101 Locke Condos is a stylish, boutique condominium development offering 84 units on six floors in the heart of Hamilton's Locke St neighbourhood.
Urban West Condos - 427 Aberdeen Avenue
Completed in 2014, Urban West Condos offers 33 units between 694 and 1,438 square feet. Located in the west end of Hamilton's core, just steps from trendy Locke Street.
Bentley Place - 67 Caroline Street South
Bentley Place Condos at 67 Caroline Street South, Hamilton offers 96 two bedroom units in a 21 floor building located in downtown Hamilton. Built in 1991, the floor plans are between 1,225 to almost 2,000 square feet.
Queen 75 - 75 Queen Street North
Queen 75 is a 24 floor condominium building located at 75 Queen Street North in Downtown Hamilton. It offers two and three bedroom floor plans.
Core Lofts - 66 Bays Street South
Core Lofts at 66 Bay Street South is a former Bell Canada office building that was converted into condominium lofts in 2005. Eight floors and 98 units between 628 and 1830 square feet.
Stinson School Lofts - 200 Stinson Street
Unique conversion of the historical Stinson School into 66 unique residential lofts. Exposed brick and ductwork, soaring ceiling and large windows with floor plans from 634 to 1950 square feet.
FilmWork Lofts - 80 King William Street
Developed in the former Hamilton Spectator printing press building, FilmWork Lofts is a 50 unit condominium offering large live-work lofts and 2-storey penthouse level lofts.
Witton Lofts - 50 Murray Street West
Witton Lofts at 50 Murray Street West is a conversion of a former 1920s school with an additional three stories of all glass units. Completed in 2013 by Core Urban Inc., it offers 36 units with floor plans between 980 and 1,450 square feet.
Chateau Royale - 135 James Street South
Luxury living in the Chateau Royale. With its central location in Downtown Hamilton, the 15-storey condo development is all about convenience. Good range of floor plan options and affordable luxury lifestyle. Popular with investors.
Cityview Terrace - 47 Caroline Street North
Located in Downtown Hamilton at the corner of Carline and Napier, this six storey, 51 unit condo building offers 16 different floor plans ranging from 648 and 1,259 square feet. Some of the fourth floor units feature large terraces.
The Olympia Condos - 150 Charlton Avenue East
Built in 1975, The Olympia Condos sit at150 Charlton Avenue in the Corktown neighbourhood. It is one of Hamilton's tallest buildings at 32 floors and offers one and two bedroom floor plans.
The Village Hill - 222 Jackson Street West
Spacious, affordable units in this 16 storey condominium building located in downtown Hamilton. Many units at the Village Hill have been updated.
The Annex Lofts - 11 Rebecca Street
The Annex Loft building is an industrial-style redevelopment of the former Eatons Annex. It offers 40 residential units with soaring ceilings, exposed duct work and polished concrete floors.
The Clifton - 81 Charlton Avenue East
The Clifton is an affordable condominium conversion in the heart of Corktown. It offers onsite laundry and units ranging from 519 to 760 square feet.
Acclamation - 181 James Street North
Completed in 2019, The Residences at Acclamation sits in the heart of the trendy James Street North arts district. This eight story, 73 unit development offers units between 670 and 1,360 square feet.


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Quick Answer: What Was The Most Expensive Super Bowl Commercial 2019?
Why do companies pay so much for Super Bowl commercials?
How much is a 30 second commercial?
What is the most expensive thing in this world?
What was the most expensive commercial?
How much did a commercial cost during the Super Bowl?
Does Budweiser sell their Clydesdales?
What is the most famous commercial of all time?
Who paid the most for a Super Bowl commercial?
Why does Budweiser use Clydesdales?
Did Budweiser have a commercial in the Super Bowl?
What was the best Super Bowl commercial?
Who is the kid in the Super Bowl Commercial 2020?
Who won Super Bowl commercial?
What was the first commercial ever?
What is the new Budweiser commercial?
What is the most expensive commercial in the Super Bowl?
What was the most popular Super Bowl commercial 2020?
How much does Pepsi pay to sponsor Super Bowl?
How much did Budweiser pay for Super Bowl ads?
How much do Super Bowl tickets cost 2020?
Who is the blonde in the new Budweiser commercial?
Why are Super Bowl commercials so expensive.
Basically, the Super Bowl is the annual event with the most eyeballs — an estimated 103 million watched last year's game.
The longer answer is that Super Bowl commercials cost a lot because they've become as much a part of the viewing experience as the game itself..
For local television stations, advertisers can expect to pay a minimum of $5 per 1,000 viewers for a 30-second commercial. Based on data provided by Adage, a 30-second spot broadcast nationally averages around $115,000 in 2019. The average cost placements for 30-second Super Bowl ads can go for upward of $5.25 million.
17 Most Expensive Things On This PlanetYacht History Supreme, 4.5 billion USD.Antilia, 1 billion USD. … 1963 Ferrari GTO, 52 million USD. … 'The Card Players' (painting), 260 million USD. … The 'Perfect Pink', 23 million USD. … Parking spot Manhattan, 1 million USD. … Feather of Huia Bird, 10,000 USD. … More items…
The Most Expensive Commercials Of All TimeCARLTON DRAUGHT: $1.5 million (2005) … FERRARI/SHELL: $4.5 million (2007) … HONDA: $6.2 million (2003) … PEPSI: $8.1 million (2002) … CHRYSLER: $12 million (2011) … AVIVA: $13.4 (2008) … GUINNESS: $16 million (2007) … CHANEL: $33 million (2004)
Since 2017, the average cost of a Super Bowl ad has continued to skyrocket. In 2018, the cost of a 30-second Super Bowl commercial was $5.2 million. In 2019, the average cost of a 30-second ad raised to $5.25 million.
With their beautiful color and enormous stature, the brewing company exploits an army of clone-like Clydesdale horses to pull turn-of-the-century beer wagons. … Budweiser then sells off "used up" Budweiser horses when they're no longer of use to the company.
#1: Apple – "1984" (1984) … #2: Wendy's – "Where's the Beef?" (1984) … #3: Tootsie Pop – "How Many Licks?" (1968) … #4: Coca-Cola – "Meet Joe Greene" (1979) … #5: Budweiser – "Whassup?!" (1999) … #6: McDonald's – "The Showdown" (1993) … #7: Energizer – "Escape of the Bunny" (1989)More items…•
$12.4 million (2011) – The most expensive ad in Super Bowl history changed the life of the brand and the city of Detroit. Seeing Eminem driving a Chrysler 200 around the city not only marked the renewal of Chrysler but also increased sales by 50% and helped energise the Motor City's auto industry.
The Budweiser Clydesdales refers to teams of Clydesdale horses used to pull restored turn-of-the-century beer wagons for Budweiser. They first appeared in 1933, given as a gift to the brewery's CEO from his son to celebrate the repeal of prohibition. … Such events aptly demonstrate the effect of using the horses.
Budweiser released its 60-second Super Bowl 2020 commercial, "Typical American," this morning ahead of the Big Game.
20 Best Super Bowl Commercials of All TimeCoca-Cola: "Hey Kid, Catch!" – Super Bowl XIV (1980)Apple: 1984 – Super Bowl XVIII. … Wendy's: "Where's the Beef?" – Super Bowl XVIII (1984) … Budweiser: Frogs – Super Bowl XXIX (1995) … McDonald's: The Showdown – Super Bowl XXVII (1993) … Pepsi: Cindy Crawford – Super Bowl XXVI (1992) … More items…•
Maxwell 'Bunchie' YoungMeet Maxwell 'Bunchie' Young, the 'Super Bowl kid' from LA who won over viewers. When Maxwell 'Bunchie' Young is not attending school in Los Angeles, the 13 year old is starring in Super Bowl commercials.
Bill MurraySuper Bowl Fanpulse: Bill Murray wins the best commercial.
The first television commercial ever aired 75 years ago today. On July 1, 1941, at 2:29 p.m., the first television commercial appeared on screens in the US. The shaky, 10-second spot for Bulova, a New York-based watch company, aired right before a Brooklyn Dodgers and Philadelphia Phillies baseball game and cost $9.
LOUIS — Budweiser is out with a new commercial that celebrates friends getting back together across the country as social distancing requirements begin to relax. The 'Reunited with Buds' commercial was created by Anomaly Canada and is set to Queen's 'Don't Stop Me Now.
By contrast, Super Bowl XLVI set what was then a record for the price of a Super Bowl advertisement, selling 58 spots (including those longer than 30 seconds) during the game, generating $75 million USD for NBC; the most expensive advertisement sold for $5.84 million.
Here are our picks for the Top 10 Ads of Super Bowl 2020:Hyundai Sonata: "Smaht Pahk"Snickers: "#SnickersFixTheWorld" … Mtn Dew Zero Sugar: "As Good as the Original" … Tide: "Super Bowl Now, Laundry Later" … Microsoft: "Be the One: Katie Sowers" … Porsche: "The Heist" … Rocket Mortgage: "Comfortable" With Jason Momoa. … More items…•
Pepsi reportedly pays over $7 million per year to sponsor the Super Bowl halftime show, and has bought rights to the show for the past 5 years. Additionally, insurance for the show costs over $100,000, according to MarketWatch.
Anheuser-Busch InBev spent an estimated $41 million. Regardless of which metric you choose, the Super Bowl is the biggest advertising event of the year with some correspondingly impressive buys by the biggest spenders.
Super Bowl LIV ticket cost: $4,220 to $60,000 And that can be pricey. The average price per ticket is about $7,000, according to StubHub's analysis of tickets resold through their site, although the cheapest seats have been trending down in the last week. As of the morning of Jan.
actress Charlize TheronThe brewer, which flooded the Super Bowl with eight ads, will appear in Sunday's Academy Awards broadcast on ABC with a 60-second spot for Budweiser starring actress Charlize Theron.
Quick Answer: What Happens If I Change Tire Sizes?
Is 70 or 75 a wider tire? 70 (or 75) is the ratio of
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Do car washes cause scratches? By Tom Fraser, 10 Jul
Question: What Size Is A 285 65r18?
How big is a 285 tire? P-Metric Tire Sizes – P-Metric
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How many miles do Jeep rubicons last? 400,000 milesA
Question: What Marilyn Monroe Movies Are On Netflix?
Are there any Marilyn Monroe movies on Netflix?


I am excited about the speaker/theologian who has been invited for this JULY by West Coast Centre for (r)Evoluitonary Theology and Vancouver School of Theology – Thomas Oord. He comes from a Progressive Evangelical background and has been a key proponent of Open and Relational Theism, working hard at articulating a new understanding of how God works in the world. Oord's thinking has actually gotten him fired from the Evangelical Seminary where he was teaching, but that seems to have spurred him on to dig ever more deeply into how one can affirm a loving and involved God while taking with the utmost seriousness, the realities of evil, pain and suffering, and the role of chance and randomness in the unfolding of reality.
If conventional notions of God make less and less sense to you, you'll find Thomas Jay Oord's new book a breath of fresh air. Simply put, "God Can't" presents an understanding of God that thoughtful, ethical people can believe in.
Victims of trauma sometimes hear theological responses that imply their suffering is somehow "God's will." A more careful theological reflection on the nature of the power of a God who is love can help. Oord gives us a clear and compelling alternative in this profoundly insightful and admirably concrete and accessible book.
So… let me encourage you to SIGN UP for the SUMMER SCHOOL COURSE at Vancouver School of Theology.
Open and Relational theologies provide winsome resources for answering big questions. This course answers two: Why doesn't God prevent suffering and evil? Can God stop climate change singlehandedly?
COST: $255 to audit; $330 for full credit.
I hope to see you out at VST, or if that's not possible, then perhaps at the Tuesday night lecture. If you have any questions, don't hesitate to get in touch with me.


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ACCEPTED

#### According to
Index Fungorum

#### Published in
Novitates Mycologicae 167 (1939)

#### Original name
Cyphella hyperici Velen.

### Remarks
null

redpajama_set_name: RedPajamaGithub


text:
Q: How to get key from the reference of a hash element suppose $my_ref = \$hash{'mary'}; #my_ref is a reference point to a hash element.
....
later on, how can I use $my_ref to retrieve the key of the hash element it point to? i.e how to get string 'mary' from $my_ref?
I ask this question because I have several groups of user name list, some user names appear in multiple groups which consumes memory. So I decide to create a common user name list, and let these groups only store the reference to the corresponding user name rather than user name.
e.g. originally,
%group1 = {'mary'=>1, 'luke'=1,'tom'=1,...}
%group2 = {'mary'=>1, 'sam'=1,'tom'=1,...}

Here you see 'mary' and 'tom' are shown in both group1 and group2 which consume memory. (note I do not care the value in this example, the value is here only because the data struct is a hash). So to reduce memory, I want to have a common list stores all user names:
%common_hash = {'mary'=>1, 'luke'=1,'tom'=1,'sam'=1...};
$ref1 = \$common_hash{'mary'};
$ref2 = \$common_hash{'luke'};
$ref3 = \$common_hash{'tom'};
$ref4 = \$common_hash{'sam'};

groups only store the reference of the hash element:
%group1 = {$ref1=>1, $ref2=1,$ref3=1,...};
%group2 = {$ref1=>1, $ref4=1,$ref3=1,...};

I think this approach can save much memory because:

*

*one user name is store in memory once not multiple times;

*groups stores reference (an integer) rather than string (in my case, the length of each user name is 30 bytes in average, while each integer is only 4 bytes (32 bit sys.) or 8 bytes (64 bit sys.)) (BTW, correct me if an integer does not use 4 bytes or 8 bytes.)

*using reference I can access user name immediately without looking for it.

But how can I get the user name from a group?
If I use @my_ref = keys %group1, I think I will get value of 'mary',but not 'mary'.
$result = $($my_ref[0]);

A: *

*A reference is not an integer; it's an SV, so it's going to be something like 24 bytes, not 4.

*Not that it matters, because you're not storing references, because hash keys are always strings. The keys of your %group1 etc. hashes are actually strings that look like "HASH(0x19838e2)", which is useless.

*Not that it matters, because Perl is smart enough to avoid wasting memory if the same strings are used as keys in multiple hashes. That's right, if you just did things the simple, obvious, sensible way, perl would use less memory than it does with the complicated thing you're trying to do.

A: Sorry, hashes don't work that way.  You aren't saving any memory by using a reference instead of a string as a hash key, and furthermore you are:

*

*making it harder to find data in the hash (it is obscured)

*getting in the way of Perl's internal hash optimizations (using a hash algorithm to provide O(1) lookup inside what is effectively a list).

In either case, the hash key is a scalar, which needs to be stored somewhere.  By using a reference as the hash key, now you not only need to store the reference in the hash, but also the value it is referencing, so you are now using more memory.
What led you to believe that you were saving memory by your, cough, novel approach? Have you run a memory profiler against different implementations?
Generally, you cannot get from a hash's value back to its key (although you could traverse the hash table linearly looking for it, if it were unique).  If you want to keep track of both a hash key and value, you need to do it yourself.  Some common approaches are:
# iterate through the table by key
foreach my $key (keys %hash)
{
     # here we have both the key and its corresponding value
     print "value at key $key is $hash{$key}\n";
}

# iterate through the table by keys and values
while (my ($key, $value) = each %hash)
{
     print "value at key $key is $value, which is the same as $hash{$key}\n";
}

Please read up on how hashes work in the manual.  You can also read about the keys and each functions.

A: A hash is a means of associating names with scalars. If you have a hash and a key, you have a scalar, not a reference to a hash bucket or anything like that.
my $value = $hash{name};

Is just a scalar.
my $ref = \$hash{name};

Is just a reference to a scalar. No more capable of containing information that allows you to back trace to a hash key than an anonymous reference can tell you what the name might be on the symbol table, or the lexical pad (without some help).

A: Try thinking about it like database tables. Have a user "table" / hash that associates a user id to information about the user, and have the other hashes use the user id, instead of the user's information.
my $userid = 5;
$user->{$groupid};
# would be the hash element for that user with a user id

You could then make your group lists use the numbers instead of names / usernames.
However, I think you are making more work for yourself than is needed. Have you actually run into a problem with this program using too much memory? Having duplicate keys is not a problem unless your keys contain very large size strings.
If you have a thousand different usernames (all 100 chars or less) and combined, there are 10,000 user / group relationships, then you only have:
100 bytes * 10,000 = 1MB
And to be honest, most names are 1/5 of that size: 200 KB
My suggestion would be to worry about this only if you have many MB of information (say 500 or more).


Here is a classic mistake made by numerous Poker gamers.
In most cases leading 3 spend with 3rd generally having to pay back again at least your buy-in. Find tournaments exactly where there are large stacks and the perform is normal pace, by this I imply Blinds improve each ten minutes or so.
The working Day The Music Died?


text:
Col3neg
Heenayaki Me Adare
Heenayaki Me Adare Episode 4
Posted by col3neg on 24 Nov 2022 (9:32 PM). Hosted on the Internet.
Heenayaki Me Adare Episode 10
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Summer Sun
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@visittenerife
Tenerife Family Holidays & Breaks
Best days out
Tenerife is one of the most popular destinations for family holidays among Brits, especially in winter and spring. Year-round sunshine, a balmy climate and great beaches are the most compelling reasons for visiting this largest of the Canary Islands, situated 320km off the coast of Africa.
But the island has many other striking attractions, not least Spain's highest peak (3718m), Mount Teide, where kids enjoy the cable-car ride and volcanic terrain. Other highlights include the family attractions Loro Parque and Siam Park, the scenery and the atmospheric old towns of the north.
Bear in mind that parts of Tenerife are very developed – although that in turn does mean there's lots of family-friendly accommodation and children's activities.
Let us tailor-make the perfect holiday for you.
Things to do with kids in Tenerife
Take a tour from either the south or the north for the Teide National Park (Parque Nacional del Teide), where you'll find Spain's highest mountain. The park itself, mostly above 2000m, is a glorious moon-like spectacle. Kids love taking the cable-car to within 200m of the summit, where, cloud permitting, you'll enjoy outstanding views, including four of the other Canary Islands. From the north, the tour goes up through the Orotava Valley, with a stop in Orotava to look around its historic centre. The tour from the south takes the Chío road between the Chinyero and Pico Viejo volcanoes. There are also two public bus routes: the 348, from Puerto de la Cruz, and line 342, from Playa de las Américas. Go early in the day to avoid crowds.
Stargaze - the Teide Observatory has state-of-the-art solar telescopes that you can peer through during a guided tour (ages 8+) or during the free summer open day with its children's activities. Those with children eight and over could also spend the night at the top of Teide, at the Altavista Refuge at 3,270m; alternatively, try the Parador de Las Canadas del Teide with its telescopes and indoor pool. Real space geeks might like to attend the summer Starmus Festival, attracting speakers as illustrious as Stephen Hawking and Richard Dawkins. Lastly, there are a planetarium and astronomy nights/camps at the Museum of Science and the Cosmos in La Laguna, which has plenty of interactives to keep the kids both entertained and informed. La Laguna itself is a UNESCO World Heritage site with well-preserved mansions from the last 400 years.
Explore the north of the island, home to the popular tourist area of Puerto de la Cruz–Valle de La Orotava. Magnificent beaches line the coast here, and you'll also find the Complejo Martiánez, an impressive complex of swimming pools and natural pools. Lively, cosmopolitan Puerto de la Cruz has a great range of hotels, restaurants and shops, while accommodation in neighbouring La Orotava and Los Realejos is mainly in rural houses. Puerto de la Cruz is home to botanical gardens as well as Loro Parque, a wildlife park with an aquarium, sharks and all kinds of large and small exotic animals. Other attractions of this area are the traditional gastronomy, outdoor activities and historic buildings to explore.
You won't go short of leisure facilities, outdoor activities, shopping or nightlife in the south with its lively — sometimes too lively – resorts. Nearly all have excellent conditions for diving, surfing, kite-surfing, windsurfing, sailing and canoeing, along with other watersports. Another highlight is the whale- and dolphin-watching trips from the main resorts to view the colonies off the south coast – Royal Delfin, approved by the Tenerife Town Council, offers free pick-up from resorts for its trips by glass-bottomed catamaran.
Europe's biggest waterpark, Siam Park in Costa Adeje near Playa de las Américas boasts state-of-the-art slides, rivers and rapids, and play areas for children. A free bus service operates from Los Cristianos, Playa de las Américas and Costa Adeje (leave before the park closes to beat the crowds to the bus at the end of the day).
Costa Adeje is the place to be for those who love their family holidays to have large doses of watery fun – it's also home to the Aqualand Aquapark, with more waterslides, rapids and lots of swimming pools. A free shuttle operates from Playa de las Américas and Los Cristianos, and it is also on other bus routes. Combined tickets are available with Jungle Park Las Águilas outside Playa de las Américas, a zoological and botanical park with more than 500 animals that you can view along a route of tunnels, hanging bridges, waterfalls, lagoons and caves, plus exotic bird shows.
@volcanoteideexperience
Family beside the cable-car at the top of Teide.
Local cuisine includes lots of seafood, tapas and paella. Specialities to look out for include Canary bananas, puchero, a saffron-infused stew with chorizo, beans, chickpeas and potatoes, and samu, a local fish baked on a bed of salt and served with small jacket potatoes and mojo, a spicy garlic sauce.
Canarian specialties and Spanish tapas are served in restaurants, though large resorts also offer familiar fast food and international fare. If you're self catering, you won't have any difficulty getting your regular products in supermarkets.
© Promotur Turismo Canarias, S.A.
Kite-surfing in the south.
When to go to Tenerife
The Canary Islands have great weather all year round – temperatures rarely rise above 25°C or drop below 17°C. High summer in Tenerife can be too hot for family holidays with very young children – avoid the midday sun.
Star-gazing in Tenerife.
Expect to pay £1,500–3,500 for four for a week, including flights and accommodation, depending on the standard of accommodation and time of departure.
Destination stats
Flying time4.25hrs All flight times are based on flights from UK London airports, to the capital or nearest destination airport.
Carbon footprint4.25 CO2 Estimated tonnes of CO2 produced for return flights for a family of four.
TimezoneGMT
CurrencyEuro
View of Teide.
Family holidays with TUI
Great offers and discounts available to Tenerife with TUI
Great value holidays
Fantastic savings on your family holiday to Tenerife with First Choice
Holiday deals in Tenerife
Choose from a wide range of accommodation at great prices when you booked with Jet2holidays
Book now with TUI and save up to £300 on summer 2021 holiday, plus COVID Cover and free changes included on new bookings*. Find out more and BOOK NOW.
Volcano view.
The Tenerife Carnival with Kids
Rhonda discovers that the world's second-biggest carnival has something for both adults and children.
Best Family Holidays with Football Classes
A hand-picked selection of hotels and resorts which offer football classes
A Review of the Fantasia Bahia Principe, Tenerife
Catherine checks out an ultra-child-friendly new resort.
15 Amazing Places to Take the Kids at Christmas
Great ideas for festive family holidays or breaks, whether you head to the beach, the ski slopes or the snowy wilds, or to a sparkling city.
A Family Break at the Holiday Village Tenerife, Canary Islands
We assess the growing appeal of all-inclusive family holidays at a First Choice property in Tenerife.
A Family Holiday at the Hotel Botanico & The Oriental Spa Garden, Tenerife
Gemma finds a hotel offering unique family activities alongside some serious parent pampering.
7 Winter Sun Family Holidays Within a 7-Hour Flight of the UK
Rhonda gives us the low-down on the best winter sun destinations with the kids without a long flight.
For regular family travel inspiration and ideas
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text:
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Meet the Volvo C40 Recharge Pure Electric SUV
Volvo's All-New Pure Electric Crossover is Coming Soon to Volvo Cars Princeton
Gasoline has carried cars far in the 130+ years since the first production automobile hit the streets. Still, the internal combustion engine's inherent efficiency limitations and collective greenhouse gas emissions have demonstrated the need for an alternative fuel source. Fortunately, the automotive landscape is evolving, with battery technologies continually improving and hybrid/EV popularity swiftly expanding with each new year. Automobility's destiny is electric, and Volvo is leading the way in the luxury segment with its Recharge lineup of plug-in hybrid (PHEV) and pure-electric (EV) cars and SUVs.
The transition to a more sustainable automotive future is a natural evolution for Volvo. And with the Recharge lineup, the automaker aims to field a lineup that is 50% EV and 50% PHEV by 2025. Volvo's initial pure-electric model was the XC40 Recharge small luxury crossover. And coming soon for 2022, the all-new Volvo C40 Recharge will be Volvo's first model not offered with a gasoline powertrain.
The 2022 C40 Recharge is scheduled to begin sales in late-2021/early-2022, but Trenton-area drivers near Morrisville, PA, and Hamilton, NJ, can reserve one for themselves today at Volvo Cars Princeton, in Lawrenceville.
Volvo C40 Electric Launch Page
Research Volvo's Commitment to Pure Electric
Shop/Buy
Shop Our Current New Volvo Electric Inventory
Shop Our Current New Volvo Hybrid Inventory
Volvo C40 Recharge: Performance, Range, and Charging
Pop the Volvo C40's front hood, and you'll find…extra storage space-great for keeping items like a charging cable. Instead, the Volvo C40 is set in motion by a pair of electric motors, one on each axle, generating a combined AWD output of 402 horsepower and 487 pound-feet of torque. And if you mash the accelerator pedal, the C40's near-instantaneous torque delivery will carry you from 0 to 60 mph in under 4.7 seconds.
To power the electric motors, the Volvo C40 employs a 75-kWh battery pack, giving the fastback crossover an estimated emissions-free driving range of 210 miles. With a Level 2 (240-volt, 11-kW) charger, the C40 Recharge can refill its battery from empty to full overnight (approx. 8 hours), adding around 25 miles of range per hour. And using 150-kW fast charging equipment, like what's often found at highway rest stops, you can add 50-60 miles per 10 minutes and charge the battery 80% in only 40 minutes.
Additionally, all 2022 Volvo pure-electric models come with 250 kWh of free charging on the Electrify America network. And when you've used the full 250 kWh within the first three years, Volvo will then cover Electrify America Pass+ membership fees for one year.
Volvo C40 Recharge: Accommodations and Versatility
Like all Volvo models, the C40 Recharge offers a luxurious interior that prioritizes refinement. The upscale cabin features a Swedish-minimalist-inspired aesthetic, with a streamlined layout, premium materials, seamlessly integrated technology, and intriguing design elements like backlit door and dash topographical panels and a standard panoramic glass sunroof. You don't even need a start button, as the C40 automatically senses when the key fob is in the car, allowing you to simply shift into drive and go. Additionally, the Volvo C40 utilizes a wide range of sustainable materials and is 100% leather-free.
The 2022 Volvo C40 Recharge provides spacious accommodations for five, with plenty of headroom and legroom no matter where you sit. Behind the rear seats, the C40 retains 14.6 cubic feet for your luggage or groceries. And with the second row folded down, the available cargo space expands to 47.4 cubic feet.
Volvo C40 Recharge: Technology and Safety
Google's innovative Android Automotive operating system replaces Volvo's Sensus system in the C40's 9-inch vertically-oriented central touchscreen. The new OS enables over-the-air updates, ensuring you always have access to the latest software. Plus, Google Assistant allows you to safely and intuitively operate many features and functions via "Hey, Google" voice commands. iPhone users needn't worry, as they're fully supported as well. The new infotainment system features the latest iteration of Apple CarPlay.
The Volvo C40 also comes standard with a full suite of advanced safety and driver-assistance technologies, including:
Pilot Assist semi-autonomous driving system with adaptive cruise control and lane-centering
Volvo City Safety collision mitigation
BLIS™ (Blind Spot Information System) with Steer Assist
Steering-adaptive Pixel LED headlights with automatic high beams
Pre-Order the 2022 Volvo C40 Recharge near Trenton, NJ, at Volvo Cars Princeton
Purchasing or leasing the all-new 2022 Volvo C40 Recharge will be completely online, and Volvo has made it easier than ever to custom-order your ideal model. For more information about the all-new Volvo C40 Recharge, available financing opportunities, and the ordering process, please reach out to Volvo Cars Princeton or visit our dealership, convenient to the Princeton, Trenton areas. We'll be happy to assist you.
Reach Out About the Volvo C40 Recharge Pure Electric SUV
2931 Brunswick Pike
Lawrenceville,	NJ	08648
Monday9:00 AM - 06:00 PM
Tuesday9:00 AM - 06:00 PM
Wednesday9:00 AM - 06:00 PM
Thursday9:00 AM - 06:00 PM
Friday9:00 AM - 06:00 PM
Saturday9:00 AM - 06:00 PM


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Cincinnati Reds Top Prospects: Ryan Hendrix
The Cincinnati Reds drafted Ryan Hendrix out of Texas A&M in the 5th round of the 2016 draft. The team took the strong armed reliever and sent him to Billings for two weeks before promoting him to Dayton to finish the year. To start the 2017 season Hendrix returned to Dayton to pitch out of the Dragons bullpen. The first week of the season went as well as could be expected. The right hander threw 4.2 innings with no earned runs, one walk and he struck out half of the batters he faced (9-of-18). It was the next week that was rough. He pitched in three games that week. A perfect inning with three strikeouts was bookended by two games in which he allowed four earned and three earned. The month finished out with 3.2 shutout innings over two games. Those two poor outings sabotaged his ERA, posting a 5.40 mark in 11.2 innings. That came along with seven walks and 22 strikeouts.
The usage for Ryan Hendrix slowed down in the first half of May. He only made four appearances, but three of them were multi-inning games. In all of them the righty was dominant. In 8.0 total innings he allowed two hits, two walks and he struck out 17 of the 28 batters faced. Over the second half of May, Hendrix made five appearances and threw 6.2 innings. He would walk just one batter and strike out 11 while posting a 2.70 ERA. It was a dominant month for the reliever, posting a 1.23 ERA in 14.2 innings with just three walks and 28 strikeouts. His walk rate dropped significantly and his strikeout rate jumped up.
To say that June got out to a good start would be an understatement. From the beginning of the month through the 16th, Ryan Hendrix threw 8.0 perfect innings over six appearances with 11 strikeouts. That earned him a promotion from Dayton to Daytona. Once in the Tortugas bullpen things didn't get out to a great start, though not entirely his fault. Hendrix was charged with three unearned runs in his first game. The next two were shutout appearances to round up the month. Between his two stops the right hander posted a 0.00 ERA in 11.1 innings. That came with just two walks, both in the final game and 14 total strikeouts.
The first half of July was an interesting mix for Ryan Hendrix. He threw 7.0 innings with a 1.29 ERA over six games. That's the good. But, he also walked five batters in that span. It did come with nine strikeouts. It was the second half of July that saw problems arise in a few areas. Over five games he would walk a batter in four of the games, and allow five runs in 5.2 innings. After two months of strong walk rates, things went backwards in July as he walked nine batters in 12.2 innings. That came with a 4.26 ERA and 14 strikeouts.
The struggles in the second half of July continued into August. Over the first two weeks of the month, Ryan Hendrix posted a 7.72 ERA with six walks and just two strikeouts in 4.2 innings. The remaining two weeks of the season went much better. In the final six appearances of the month, Hendrix allowed just one run over 7.0 innings. That came with two walks and just eight strikeouts. In total, he posted a 3.86 ERA in August over 11.2 innings with eight walks and 10 strikeouts.
For all 2017 Season Reviews and Scouting Reports – click here (these will come out during the week throughout the offseason).
Ryan Hendrix Scouting Report
Fastball | His fastball will work in the 93-96 MPH range on most nights and has touched 100 in the past.
Curveball | The pitch is a plus offering, sitting in the mid-80's. When at its best it's good nice, late breaking 12-6 action.
Change up | A clear third pitch, it's a fringy pitch that isn't gone too often.
When he's on top of his game, Ryan Hendrix looks like a future closer, throwing two plus pitches in the zone with his fastball and curveball. At other times, though, he will struggle with his ability to find the strikezone, and we saw that for a few stretches during the 2017 season. That did show up more in Daytona where he had 19 walks in 27.2 innings. That rate is going to have to improve as he moves forward. In 2018 if his control is more like it showed in Dayton than in Daytona, he could move up the ladder very quickly. But if things are more like the second half in Daytona in terms of control, it will be a slower climb up the ladder for Hendrix.
2018 Top Prospect List Scouting ReportsRyan Hendrix
The Unnamed Cincinnati Reds Podcast: Episode 9
Friday afternoon Reds news and notes
Reds prospect Ryan Hendrix shows why hitting is incredibly difficult
Cincinnati Reds #16 Prospect: Ryan Hendrix


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Watchdog okays building greenhouses on rare dunes next to Israel's border with Gaza – Haaretz
Posted by Editor on Monday, June 3rd 2019
The dunes near Netiv Ha'asara, which cover some 6,000 dunams, is one of the last large dune areas in the coastal plain region
Zafrir Rinat | May 29, 2019
The Israel Nature and Parks Authority has dropped its objection to allowing ecologically significant dunes near Moshav Netiv Ha'asara in the northwestern Negev to be used for greenhouses.
The decision makes the plan more likely to be approved; however, the Society for the Protection of Nature in Israel still objects to the plan, and wants an alternative site for the greenhouses to be found.
The District Planning and Building Committee has recently been discussing a plan to earmark 800 dunams (200 acres) of dunes for Netiv Ha'asara's agriculture. There is also a plan to turn much of the dunes into a forest, the planting of which could change the area's character and damage biodiversity. A number of groups have filed objections, which the committee is to discuss soon, ahead of its final decision.
The dunes near Netiv Ha'asara, which cover some 6,000 dunams, is one of the last large dune areas in the coastal plain region. They are home to herds of gazelles and rodents unique to the Eastern Mediterranean, including Anderson's gerbil and the jerboa. It is almost the only natural area on the coast where the gray monitor lizard, the largest lizard species in Israel, can be found. All in all, 15 species of endangered vertebrates inhabit the area, which puts it on the list of the 10 sites in Israel with the highest number of endangered species.
The parks authority has been able to have the area intended for a forest to be declared a nature reserve. But it had also originally opposed the greenhouses on the dunes. In its statement to the committee, it said: "Despite the distance from urban centers, this habitat is in real danger of extinction and the desire to allocate part of it for agriculture certainly does not benefit it, but rather expedites its extinction."
However, about 10 days ago the director of the authority's central district, Menahem Fried, informed the committee that the agency was lifting its objection. The decision was made after meetings between the Agriculture Ministry and the nature and parks authority.
The authority stated: "When we submitted our objection we thought the community would receive another solution for its agriculture needs, but that is not the case. This is a community in a special place on the Gaza border, which makes it living from a very specific type of agriculture. We decided to place the emphasis on declaring the rest of the area a nature reserve as an effective response in terms of managing and protecting the area. As for the agricultural area, we demanded that the plan be carried out in stages and, in keeping with a plan, that it will indeed be used for greenhouses. It should be noted that in the case of the Nitzanim Dunes farther north, compromises were also made; alongside a nature reserve, areas were earmarked for a hotel business and tourism. We try to understand national needs and find the appropriate balances."
The secretary of Netiv Ha'asara, Karen Ifargan, recently approached the committee in the Agriculture Ministry responsible for allocating land for cultivation, noting that the moshav had been given land farther away, but that it was not enough to meet the residents' needs.
"The agricultural activity in the moshav is intensive flower growing, nurseries and in particular the raising of vegetables for seeds," Ifargan said. "This activity requires that the areas be close to the community, both in terms of operations and agronomy."
Other ecological groups, represented on the committee by Shay Tahnai of the SPNI, still object to the plan. In their objection, Tahnai wrote: "A look at the areas at the disposal of Netiv Ha'asara shows many unused areas and that many greenhouses are neglected and do not serve their purpose." The SPNI also said that alternatives could be found suitable for the greenhouses. "It is unreasonable to cut into the dunes, whose ecosystems are a rare public resource, for the sake of greenhouses," the Tahnai wrote. The ecosystems can't exist in any other location, Tahnai added, while the greenhouses can.
https://www.haaretz.com/israel-news/watchdog-okays-building-greenhouses-on-rare-dunes-next-to-israel-s-border-with-gaza-1.7301862
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