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

Riemannian optimization concerns the optimization of a function defined over a Riemannian manifold. It is motivated by constrained problems that can be naturally expressed on Riemannian manifolds allowing to exploit the geometric structure of the problem and effectively transforming it into an unconstrained one. Moreover, there are problems that are not convex in the Euclidean setting, but that when posed as problems over a manifold with the right metric, are convex when restricted to every geodesic, and this allows for fast optimization \citep{neto2006convex, bento2012subgradient, bento2015proximal, allen2018operator}. That is, they are geodesically convex (g-convex) problems, cf. \cref{def:g-convex_smooth}. Some applications of Riemannian optimization in machine learning robust covariance estimation in Gaussian distributions \citep{wiesel2012geodesic}, Gaussian mixture models \citep{hosseini2015matrix}, operator scaling \citep{allen2018operator}, computation of Brascamp-Lieb constants \citep{bennett2008brascamp}, Karcher mean \citep{zhang2016fast}, Wasserstein Barycenters \citep{weber2017frank}, include dictionary learning \citep{cherian2016riemannian,sun2016complete}, low-rank matrix completion \citep{DBLP:journals/siamsc/CambierA16,heidel2018riemannian,mishra2014r3mc,tan2014riemannian,vandereycken2013low}, optimization under orthogonality constraints \citep{edelman1998geometry,DBLP:conf/icml/CasadoM19}, and sparse principal component analysis \citep{genicot2015weakly,huang2019riemannian,jolliffe2003modified}. The first seven problems are defined  over Hadamard manifolds, which we consider in this work, and the first six are g-convex problems to which our results can be applied. In fact, the optimization in these cases is over symmetric spaces, which satisfy a property that one instance of our algorithm requires, cf. \cref{thm:instantiation}.

Riemannian optimization, whether under g-convexity or not, is an extensive and active area of research, for which one aspires to develop Riemannian optimization algorithms that share analogous properties to the more broadly studied Euclidean methods, such as the following kinds of Riemannian first-order methods: deterministic \citep{bento2017iteration,wei2016guarantees,zhang2016first}, adaptive \citep{kasai2019riemannian}, projection-free \citep{weber2017frank,weber2019nonconvex}, saddle-point-escaping \citep{criscitiello2019efficiently,sun2019escaping,zhou2019faster, criscitiello2020accelerated}, stochastic \citep{hosseini2019alternative,khuzani2017stochastic,tripuraneni2018averaging}, variance-reduced \citep{sato2017riemannian,kasai2018riemannian,zhang2016fast}, and min-max methods \citep{zhang2022minimax, jordan2022first}, among others.

Riemannian generalizations to accelerated convex optimization are appealing due to their better convergence rates with respect to unaccelerated methods, specially in ill-conditioned problems. Acceleration in Euclidean convex optimization is a concept that has been broadly explored and has provided many different fast algorithms. A paradigmatic example is Nesterov's Accelerated Gradient Descent (\newtarget{def:acronym_accelerated_gradient_descent}{\newlink{def:acronym_accelerated_gradient_descent}{AGD}{}}), cf. \citep{nesterov1983method}, which is considered the first general accelerated method, where the conjugate gradients method can be seen as an accelerated predecessor in a more limited scope \citep{martinez2021acceleration}. There have been recent efforts to better understand this phenomenon in the Euclidean case \citep{allen2014linear,su2014differential,drori2014performance,wibisono2016variational, diakonikolas2017approximate,joulani2020simpler}, which have yielded some fruitful techniques for the general development of methods and analyses. These techniques have allowed for a considerable number of new results going beyond the standard oracle model, convexity, or beyond first-order, in a wide variety of settings \citep{tseng2008accelerated, beck2009fista,wang2015unified,DBLP:conf/stoc/ZhuO15,allen2016katyusha, allen2017natasha, carmon2017convex, diakonikolas2017accelerated, hinder2019near, DBLP:conf/colt/GasnikovDGVSU0W19, ivanova2021adaptive, kamzolov2020near, criado2021fast},

among many others. There have been some efforts to achieve acceleration for Riemannian algorithms as generalizations of \newlink{def:acronym_accelerated_gradient_descent}{AGD}{}, cf. \cref{sec:related_work}. These works try to answer the following fundamental question:

\begin{center}
\textit{Can a Riemannian first-order method enjoy the same rates of convergence as Euclidean \newlink{def:acronym_accelerated_gradient_descent}{AGD}{}?}
\end{center}

The question is posed under (possibly strongly) geodesic convexity and smoothness of the function to be optimized. And due to the lower bound in \citep{criscitiello2022negative}, we know the optimization must be under bounded geodesic curvature of the Riemannian manifold, and we might have to optimize over a bounded domain.

\paragraph{Main results} In this work, we study the question above in the case of Hadamard manifolds $\M$ of bounded sectional curvature and provide an instance of our framework for a wide class of Hadamard manifolds. For a differentiable $\f:\M\to\R$ with a global minimizer at $\xastg$, let $\xInit \in \M$ be an initial point and $\RR$ be an upper bound on the distance $\newlink{def:distance}{d}(\xInit, \xastg)$. If $f$ is $L$-smooth and (possibly $\mu$-strongly) g-convex in a closed ball of center $\xastg$ and radius $\bigo{\RR}$, our algorithms obtain the same rates of convergence as \newlink{def:acronym_accelerated_gradient_descent}{AGD}{}, up to logarithmic factors and up to a geometric penalty factor, cf. \cref{thm:instantiation}. See \cref{table:comparisons:riemannian} for a succint comparison among accelerated algorithm and their rates. This algorithm is a consequence of the general framework we design:

\emph{Riemacon: A general accelerated scheme.} We design a Riemannian accelerated inexact proximal point method that enjoys the same rates as the Euclidean accelerated proximal point method when approximating $\min_{x\in\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}}\f(x)$, up to logarithmic factors and up to a geometric penalty factor, where $\f :\NN \subset \M  \to \R$ is a  g-convex (or strongly g-convex) differentiable function in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}} \subset\NN$, cf. \cref{thm:g_convex_acceleration}. Note $\f$ does not need to be smooth, provided access to the inexact prox step.

For smooth functions, we show that with access to a (not necessarily accelerated) constrained linear subroutine for strongly g-convex and smooth problems, we can inexactly solve the proximal subproblem from a warm-start point to enough accuracy so it can be used in our accelerated outer loop, in the spirit of other Euclidean algorithms like Catalyst \citep{lin2017catalyst}. After building this machinery, we show that we are able to implement an inexact ball optimization oracle, cf. \citep{carmon2020acceleration}, as an instance of our solution. Crucially, the diameter $\D$ of this ball depends on $\RR$ and the geometry only, so in particular it is independent on the condition number of $\f$. We can use the linearly convergent algorithm in \citep{criscitiello2022negative} for the implementation of the prox subroutine and we show that iterating the application of the ball optimization oracle leads to global accelerated convergence. We also define a convex Euclidean projection oracle, and show that it allows to implement another subroutine that enjoys better geometric penalties.

Importantly, our algorithms obtain acceleration without an undesirable assumption that most previous works had to make: that the iterates of the algorithm stay inside of a pre-specified compact set without any mechanism for enforcing or guarateeing this condition. This is not the same as assuming the iterates are bounded in \emph{some} compact set, see \hyperlink{sec:handling_constraints_to_bound_geometric_penalties}{this discussion}. All methods require some constraints to bound geometric penalties but to the best of our knowledge only two previous methods are able to enforce these constraints, and they apply to the limited settings of local optimization \citep{criscitiello2022negative} and constant sectional curvature manifolds \citep{martinez2021acceleration}, respectively. Techniques in the rest of papers resort to just assuming that the iterates of their algorithms are always feasible. Removing this condition in general, global, and fully accelerated methods was posed as an open question in \citep{kim2022accelerated}, that we solve for a wide class of Hadamard manifolds. The difficulty of constraining problems in order to bound geometric penalties as well as the necessity of achieving this goal in order to provide full optimization guarantees with bounded geometric penalties is something that has also been noted in other kinds of Riemannian algorithms, cf. \citep{hosseini2020recent}.

The question concerning whether there are Riemannian analogs to Nesterov's algorithm that enjoy similar rates is a question that, to the best of our knowledge, was first formulated in \citep{zhang2016first}. In particular, since Nesterov's \newlink{def:acronym_accelerated_gradient_descent}{AGD}{} uses a proximal operator of a function's linearization, they ask whether there is a Riemannian analog to this operation that could be used to obtain accelerated rates in the Riemannian case.

We show that, instead, a proximal step with respect to the \textit{whole} function can be approximated efficiently in Hadamard manifolds and it can be used along with an accelerated outer loop.
Previously known Riemannian proximal methods either obtain asymptotic analyses, assume exact proximal computation, or work with approximate proximal operators by using different inexactness conditions as ours, and none of them show how to implement the proximal operators or obtain accelerated proximal point methods, cf. \cref{sec:related_work}.

\begin{table}[!ht]
    \centering
    \caption{Convergence rates of related works with provable guarantees for smooth problems over uniquely geodesic manifolds. Column \textbf{K?} $\rightarrow$ sectional curvature?, \textbf{G?} $\rightarrow$ global algorithm?: any initial distance to a minimizer is allowed. Here L and L$'$ mean they are local algorithms that require initial distance $\bigo{(\L /\mu)^{-3/4}}$ and $\bigo{(\L /\mu)^{-1/2}}$, respectively. Column \textbf{F?} $\rightarrow$ full acceleration?: dependence on $\L $, $\mu$, and $\epsilon$ like  AGD  up to possibly log factors. Column \textbf{C?} can enforce some constraints?: All methods require their iterates to be in some \textbf{pre-specified} compact set, but works with {\color{red} \xmark}{} just assume the iterates will remain within the constraints. We use ${\protect\W} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \sqrt{\frac{\L }{\mu}}\log(\frac{\L \RR^2}{\epsilon})$. $^*$A mild condition on the covariant derivative of the metric tensor is required, cf. \cref{assump:bounded_curvature_tensor}, \cref{sec:proofs_ball_oracle}. $^{**}$With access to the convex projection oracle in \cref{sec:other_subroutine}. }
    \label{table:comparisons:riemannian}
\begin{tabular}{lllccccc}
    \toprule
    \textbf{Method}   &  \textbf{g-convex} & \textbf{$\mu$-st. g-cvx} & \textbf{K?} & \textbf{G?} & \textbf{F?}& \textbf{C?}  \\
    \midrule
    \midrule
    \citep[\newlink{def:acronym_accelerated_gradient_descent}{AGD}{}]{nesterov2005smooth}                 & $O(\sqrt{\L\RR^2/\epsilon})$  & $\bigo{\W}$ & $0$ & {\color{green} \cmark} & {\color{green} \cmark} & {\color{green} \cmark} \\
    \midrule
    \citep{zhang2018towards} & -  & $\bigo{\W}$ &  {\footnotesize bounded} & L  & {\color{green} \cmark} & {\color{red} \xmark}   \\
    \citep{ahn2020nesterov}                          & - & $\bigotilde{\frac{\L}{\mu} + \W}$ &  {\footnotesize bounded} & {\color{green} \cmark} & {\color{red} \xmark} & {\color{red} \xmark}  \\

    \citep{martinez2020global}  & $\bigotilde{\zetar^{\frac{3}{2}}\sqrt{\zetar+\L\RR^2/\epsilon}}$ & $\bigotilde{\zetar^{\frac{3}{2}}\cdot\W}$ & {\footnotesize ctant.$\neq 0$} & {\color{green} \cmark} & {\color{green} \cmark} & {\color{green} \cmark}  \\
    \citep{criscitiello2022negative}  & -  & $\bigo{\W}$ &  {\footnotesize bounded$^*$} & L$'$ & {\color{green} \cmark} & {\color{green} \cmark}   \\
    \citep{kim2022accelerated}  & $\bigo{\zetar \sqrt{\L \RR^2/\epsilon}}$ & $\bigo{\zetar \cdot \W}$ & {\footnotesize bounded} & {\color{green} \cmark} & {\color{green} \cmark} & {\color{red} \xmark}  \\

    \textbf{Theorem \ref{thm:instantiation}}  & ${\bigotilde{\zetar^2 \sqrt{\zetar+\L\RR^2/\epsilon}}}$ & $\bigotilde{\zetar^2\cdot \W}$ & {\footnotesize Hadamard$^*$  } & {\color{green} \cmark} & {\color{green} \cmark} & {\color{green} \cmark} \\
    \textbf{Appendix \ref{sec:other_subroutine}}$^{**}$  & ${\bigotilde{\zetar \sqrt{\zetar+\L\RR^2/\epsilon}}}$ & $\bigotilde{\zetar\cdot \W}$ & {\footnotesize Hadamard$^*$  } & {\color{green} \cmark} & {\color{green} \cmark} & {\color{green} \cmark} \\
    \bottomrule
\end{tabular}
\end{table}

\subsection{Preliminaries} \label{sec:preliminaries}

We provide definitions of Riemannian geometry concepts that we use in this work. The interested reader can refer to \citep{petersen2006riemannian, bacak2014convex} for an in-depth review of this topic, but for this work the following notions will be enough. A Riemannian manifold $(\MOnly,\mathfrak{g})$ is a real $C^{\infty}$ manifold $\MOnly$ equipped with a metric $\mathfrak{g}$, which is a smoothly varying, i.e., $C^{\infty}$, inner product. For $x \in \MOnly$, denote by $\newtarget{def:tangent_space}{\Tansp{x}}\MOnly$ the tangent space of $\MOnly$ at $x$. For vectors $v,w  \in \Tansp{x}\MOnly$, we denote the inner product of the metric by $\innp{v,w}_x$ and the norm it induces by $\norm{v}_x \stackrel{\mathrm{\scriptscriptstyle def}}{=} \sqrt{\innp{v,v}_x}$. Most of the time, the point $x$ is known from context, in which case we write $\innp{v,w}$ or $\norm{v}$.

A geodesic of length $\ell$ is a curve $\gamma : [0,\ell] \to \MOnly$ of unit speed that is locally distance minimizing. A uniquely geodesic space is a space such that for every two points there is one and only one geodesic that joins them. In such a case the exponential map $\newtarget{def:riemannian_exponential_map}{\expon{x}} : \Tansp{x}\MOnly\to \MOnly$ and the inverse exponential map $\exponinv{x}:\MOnly\to \Tansp{x}\MOnly$ are well defined for every pair of points, and are as follows. Given $x, y\in\MOnly$, $v\in \Tansp{x}\MOnly$, and a geodesic $\gamma$ of length $\norm{v}$ such that $\gamma(0) =x$, $\gamma(\norm{v})=y$, $\gamma'(0)=v/\norm{v}$, we have that $\expon{x}(v) = y$ and $\exponinv{x}(y) = v$. We denote by $\newtarget{def:distance}{\newlink{def:distance}{d}}(x,y)$ the distance between $x$ and $y$, and note that it takes the same value as $\norm{\exponinv{x}(y)}$. The manifold $\MOnly$ comes with a natural parallel transport of vectors between tangent spaces, that formally is defined from a way of identifying nearby tangent spaces, known as the Levi-Civita connection $\nabla$ \citep{levi1977absolute}. We use this parallel transport throughout this work. As all previous accelerated related works do, discussed in \cref{sec:related_work}, we assume that we can compute the exponential and inverse exponential maps, and parallel transport of vectors for our manifold.

Given a $2$-dimensional subspace $V \subseteq \Tansp{x}\MOnly$ of the tangent space of a point $x$, the sectional curvature at $x$ with respect to $V$ is defined as the Gauss curvature, for the surface $\expon{x}(V)$ at $x$. The Gauss curvature at a point $x$ can be defined as the product of the maximum and minimum curvatures of the curves resulting from intersecting the surface with planes that are normal to the surface at $x$. A Hadamard manifold is a complete simply connected Riemannian manifold whose sectional curvature is non-positive, like the hyperbolic space or the space of $n\times n$ symmetric positive definite matrices with the metric $\innp{X, Y}_{A} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \operatorname{Tr}(A^{-1}XA^{-1}Y)$ where $X, Y$ are in the tangent space of $A$. Hadamard manifolds are uniquely geodesic. Note that in a general manifold $\expon{x}(\cdot)$ might not be defined for each $v\in \Tansp{x}\MOnly$, but in a Hadamard manifold of dimension $n$, the exponential map at any point is a global diffeomorphism between $\Tansp{x}\MOnly\cong\R^n$ and the manifold, and so the exponential map is defined everywhere. We now proceed to define the main properties that will be assumed on our model for the function to be minimized and on the feasible set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$.

\begin{definition}[Geodesic Convexity and Smoothness] \label{def:g-convex_smooth}
    Let $\f:\NN\subset \MOnly \to \R$ be a differentiable function defined on an open set $\NN$ contained in a Riemannian manifold $\MOnly$. Given $\newtarget{def:riemannian_smoothness_of_F}{\L}\geq \newtarget{def:strong_g_convexity_of_F}{\mu} > 0$, we say that $\f$ is \emph{$\L$-smooth} in a set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}} \subseteq \NN$ if for any two points $x, y \in \newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$, $\f$ satisfies
    \[
        \f(y) \leq \f(x) + \innp{\nabla \f(x), \exponinv{x}(y)} + \frac{\L}{2}\newlink{def:distance}{d}(x,y)^2.
    \]

    Analogously, we say that $\f$ is \emph{$\mu$-strongly g-convex} in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$, if for any two points $x, y \in \newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$, we have
    \[
    \f(y) \geq \f(x) + \innp{\nabla \f(x), \exponinv{x}(y)} + \frac{\mu}{2}\newlink{def:distance}{d}(x,y)^2.
    \]
    If the previous inequality is satisfied with $\mu=0$, we say the function is \emph{g-convex} in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$.
\end{definition}

We present the following fact about the squared-distance function, when one of the arguments is fixed. The constants $\zeta{\D}$, $\delta{\D}$ below appear everywhere in Riemannian optimization because, among other things, \cref{fact:hessian_of_riemannian_squared_distance} yields Riemannian inequalities that are analogous to the equality in the Euclidean cosine law of a triangle, cf. \cref{lemma:cosine_law_riemannian}, and these inequalities have wide applicability in the analyses of Riemannian methods.

\begin{fact}[Local information of the squared-distance]\label{fact:hessian_of_riemannian_squared_distance}
    Let $\MOnly$ be a Riemannian manifold of sectional curvature bounded by $[\newtarget{def:minimum_sectional_curvature}{\kmin}, \newtarget{def:maximum_sectional_curvature}{\kmax}]$ that contains a uniquely g-convex set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}\subset \MOnly$ of diameter $\D<\infty$. Then, given $x,y\in\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ we have the following for the function $\newtarget{def:squared_distance_function_div_by_2}{\Phi_x}:\M \to\R$, $y \mapsto \frac{1}{2}\newlink{def:distance}{d}(x, y)^2$:
\[
    \nabla \Phi_x(y) = -\exponinv{y}(x)\text{\quad \quad and \quad \quad} \delta{\D}\norm{v}^2 \leq \Hess  \Phi_x(y)[v, v] \leq \zeta{\D} \norm{v}^2,
\]
These bounds are tight for spaces of constant sectional curvature. The geometric constants are
\begin{equation*}\label{eq:defi_zeta_and_delta}
    \newtarget{def:zeta}{\zeta{\D}} \stackrel{\mathrm{\scriptscriptstyle def}}{=}
        \begin{cases}
            \D\sqrt{\abs{\kmin}}\coth(\D\sqrt{\abs{\kmin}})& \text{ if } \kmin \leq 0 \\
            1 &  \text{ if } \kmin > 0\\
        \end{cases}%
        ,
\end{equation*}
and
\begin{equation*}\label{eq:defi_delta}
        \newtarget{def:delta}{\delta{\D}} \stackrel{\mathrm{\scriptscriptstyle def}}{=}
        \begin{cases}
            1 & \text{ if }\kmax \leq 0\\
            \D\sqrt{\kmax}\cot(\D\sqrt{\kmax})& \text{ if } \kmax > 0 \\
        \end{cases}.
\end{equation*}
    Consequently, $\Phi_x$ is $\delta{\D}$-strongly g-convex and $\zeta{\D}$-smooth in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$. See \citep{kim2022accelerated}, for instance.
In particular, for Hadamard manifolds, $\Phi_x$ is $1$-strongly g-convex and sublevel sets of g-convex functions are g-convex sets, so balls are g-convex in these manifolds \citep{bacak2014convex}.
\end{fact}

\subsection{Notation}\label{sec:notation} Let $\newtarget{def:M_manifold}{\M}$ be a uniquely geodesic $\newtarget{def:dimension}{\n}$-dimensional Riemannian manifold. Given points $x, y, z\in \M$, we abuse the notation and write $y$ in non-ambiguous and well-defined contexts in which we should write $\exponinv{x}(y)$. For example, for $v\in \Tansp{x}\M$ we have $\innp{v, y - x} = -\innp{v, x - y} = \innp{v, \exponinv{x}(y)- \exponinv{x}(x)} = \innp{v, \exponinv{x}(y)}$; $\norm{v-y} = \norm{v-\exponinv{x}(y)}$; $\norm{z-y}_x = \norm{\exponinv{x}(z)-\exponinv{x}(y)}$; and $\norm{y-x}_x =\norm{\exponinv{x}(y)} =\newlink{def:distance}{d}(y, x)$. We denote by $\newtarget{def:geodesically_convex_feasible_region}{\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}}$ a compact, uniquely geodesic g-convex set of diameter $\newtarget{def:diameter_of_geodesically_convex_feasible_region}{\D}$ contained in an open set $\newtarget{def:N_open_set_in_manifold}{\NN} \subset\M$ and we use $\newtarget{def:indicator_function}{\indicator{\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}}}$ for the indicator function of $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$, which is $0$ at points in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ and $+\infty$ otherwise. For a vector $v\in\Tansp{y}\M$, we use $\newtarget{def:parallel_transport}{\Gamma{y}{x}}(v) \in \Tansp{x}\M$ to denote the parallel transport of $v$ from $\Tansp{y}\M$ to $\Tansp{x}\M$ along the unique geodesic that connects $y$ to $x$. We call $\newtarget{def:riemannian_function_f}{\f}:\NN \subset \M\to\R$ a differentiable $\L $-smooth g-convex function we want to optimize. We use $\newtarget{def:accuracy_epsilon}{\epsilon}$ to denote the approximation accuracy parameter, $\xk[0] \in \newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ for the initial point of our algorithms, and $\newtarget{def:initial_distance_for_X}{\RInit} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \newlink{def:distance}{d}(\xk[0], \xast)$ for the initial distance to an arbitrary constrained minimizer $\newtarget{def:optimizer_at_X}{\xast} \in \argmin_{x\in\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}}\f(x)$. We use $\newtarget{def:distance_or_bound_to_global_minimizer}{\RR}$ for an upper bound on the initial distance $\newlink{def:distance}{d}(\xInit,\xastg)$ to an unconstrained minimizer $\newtarget{def:global_optimizer}{\xastg}$, if it exists. The big-$O$ notation $\newtarget{def:big_o_tilde}{\bigotilde{\cdot}}$ omits $\log$ factors.
Note that in the setting of Hadamard manifolds, the bounds on the sectional curvature are $\kmin \leq \kmax \leq 0$. Hence for notational convenience, we define $\newtarget{def:zeta_without_subindex_which_means_zeta_D}{\zetad} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \zeta{\D} = \D\sqrt{\abs{\kmin}}\coth(\D\sqrt{\abs{\kmin}}) \geq 1$, $\newtarget{def:delta_without_subindex_which_means_zeta_D}{\deltad} \stackrel{\mathrm{\scriptscriptstyle def}}{=} 1$, and similarly $\newtarget{def:zeta_without_subindex_which_means_zeta_RR}{\zetar}\stackrel{\mathrm{\scriptscriptstyle def}}{=} \zeta{\RR}$ and $\newtarget{def:delta_without_subindex_which_means_zeta_RR}{\deltar}\stackrel{\mathrm{\scriptscriptstyle def}}{=}\delta{\RR}=1$.  If $v\in \Tansp{x}\M$, we use $\newtarget{def:euclidean_projection}{\Pii}{\newtarget{def:closed_ball}{\ball}(0, r)}(v) \in \Tansp{x}\M$ for the projection of $v$ onto the closed ball with center at $0$ and radius $r$.

\section{Algorithmic framework and pseudocode}\label{sec:algorithm}
In this section, we present our \textbf{Riema}nnian \textbf{ac}celerated algorithm for \textbf{con}strained g-convex optimization, or \newtarget{def:Riemacon}{\newlink{def:Riemacon}{Riemacon}{}}\footnote{Riemacon rhymes with ``rima con'' in Spanish.}. This is a general framework that we later instantiate to provide a full algorithm. Recall our abuse of notation for points $p \in\M$ to mean $\exponinv{q}(p)$ in contexts in which one should place a vector in $\Tansp{q}\M$ and note that in our algorithm $\xk$ and $\yk$ are points in $\M$ whereas $\zx \in \Tansp{\xk}\M, \zy, \zybar  \in \Tansp{\yk}\M$.

We start with an interpretation of our algorithm that helps understanding its high-level ideas. The following intends to be a qualitative explanation, and we refer to the pseudocode and the appendix for the exact descriptions and analysis. Euclidean accelerated algorithms can be interpreted, cf. \citep{allen2014linear}, as a combination of a gradient descent (\newtarget{def:acronym_gradient_descent}{\newlink{def:acronym_gradient_descent}{GD}{}}) algorithm and an online learning algorithm with losses being the affine lower bounds $\f(\xk) + \innp{\nabla \f(\xk), \cdot-\xk}$ we obtain on $\f(\cdot)$ by applying convexity at some points $\xk$. That is, the latter builds a lower bound estimation on $\f$. By selecting the next query to the gradient oracle as a cleverly picked convex combination of the predictions given by these two algorithms, one can show that the instantaneous regret of the online learning algorithm can be compensated by the local progress \newlink{def:acronym_gradient_descent}{GD}{} makes, up to a difference of potential functions, which leads to accelerated convergence. In Riemannian optimization, there are two main obstacles. Firstly, the first-order approximations of $\f$ at points $x_k$ yield functions that are affine but only with respect to their respective $\Tansp{x_k}\M$, and so combining these lower bounds that are only simple in their tangent spaces makes obtaining good global estimations not simple. Secondly, when one obtains such global estimations, then one naturally incurs an instantaneous regret that is worse by a factor than is usual in Euclidean acceleration. This factor is a geometric constant depending on the diameter $\D$ of a set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ where the iterates and a (possibly constrained) minimizer lie. As a consequence, the learning rate of \newlink{def:acronym_gradient_descent}{GD}{} would need to be multiplicatively increased by such a constant with respect to the one of the online learning algorithm in order for the regret to still be compensated with the local progress of \newlink{def:acronym_gradient_descent}{GD}{} (and the rates worsen by this constant). But if we fix some $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ of finite diameter, because \newlink{def:acronym_gradient_descent}{GD}{}'s learning rate is now larger, it is not clear how to keep the iterates in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$. And if we do not have the iterates in one such set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$, then our geometric penalties could grow arbitrarily.

\begin{algorithm}[!ht]
    \caption{Riemacon: \textbf{Riema}nnian \textbf{Ac}celeration - \textbf{Con}strained g-Convex Optimization}
    \label{alg:accelerated_gconvex}

\begin{algorithmic}[1]
    \REQUIRE Feasible set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$. Initial point $\newtarget{def:initial_point}{\xInit} \in\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}\subset\NN$. Diff. function $\f:\NN\subset\M\to\R$ for a Hadamard manifold $\M$ that is g-convex in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$. Optionally: final iteration $T$ or accuracy $\epsilon$. If $\epsilon$ is provided, compute the corresponding $T$, cf. \cref{thm:g_convex_acceleration}.

    \paragraph{Parameters:}
    \begin{itemize}
        \item Geometric penalty $\newtarget{def:extra_geom_penalty_xi}{\xi} \stackrel{\mathrm{\scriptscriptstyle def}}{=} 4\zeta{2\D} -3 \leq 8\zetad -3 =O(\zetad)$.
        \item Implicit Gradient Descent learning rate $\newtarget{def:implicit_GD_learning_rate_lambda}{\lambda} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \zeta{2\D}/\L $.
        \item Mirror Descent learning rates $\newtarget{def:MD_step_length}{\etak} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \ak/\xi$.
        \item Proportionality constant in the proximal subproblem accuracies: $\newtarget{def:loss_factor_in_Lyapunov}{\Deltak} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \frac{1}{(k+1)^2}$.
    \end{itemize}
    \paragraph{Definition:} (computation of this value is not needed)
    \begin{itemize}
        \item Prox. accuracies: $\newtarget{def:accuracy_of_prox_subproblems}{\sigmak} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \frac{\Deltak \newlink{def:distance}{d}(\xk, \ykast)^2}{78\lambda}$ where $\newtarget{def:optimal_proximal_operator}{\ykast} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \argmin_{y\in\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}}\{\f(y) + \frac{1}{2\lambda}\newlink{def:distance}{d}(\xk, y)^2 \}$.
    \end{itemize}

    \vspace{0.1cm}
    \hrule
    \vspace{0.1cm}
    \State $\yk[0] \gets \sigmak[0]$-minimizer of the proximal problem $\min_{y\in\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}}\{\f(y) + \frac{1}{2\lambda}\newlink{def:distance}{d}(\xInit, y)^2\}$

    \State $\zx[0][0] \gets 0 \in \Tansp{\xInit}\M$;\quad $\zybar[0][0] \gets \zy[0][0] \gets 0 \in \Tansp{\yk[0]}\M$; \quad$\Ak[0] \gets 200\lambda\xi$
    \FOR {$k = 1 \textbf{ to } T$}
        \State $\newtarget{def:discrete_step}{\ak[k]} \gets 2\lambda\frac{k+32\xi}{5}$
        \State $\newtarget{def:integral_of_steps}{\Ak[k]} \gets \ak[k]/\xi + \Ak[k-1] = \sum_{i=1}^k \ak[i]/\xi + \Ak[0] = \lambda \left(\frac{k(k+1+64\xi)}{5\xi} + 200\xi\right)$
        \State $\newtarget{def:iterate_x}{\xk} \gets \expon{\yk[k-1]}(\frac{\ak}{\Ak[k-1] + \ak} \zybar[k-1][k-1] + \frac{\Ak[k-1]}{\Ak[k-1] + \ak}\yk[k-1])=  \expon{\yk[k-1]}(\frac{\ak}{\Ak[k-1] + \ak}\zybar[k-1][k-1]) $\Comment{Coupling}

        \State $\newtarget{def:iterate_y}{\yk} \gets$ $\sigmak$-minimizer of $y\mapsto\{\f(y) + \frac{1}{2\lambda}\newlink{def:distance}{d}(\xk, y)^2\}$ in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ \label{line:subroutine}\Comment{Approximate implicit \newlink{def:acronym_riemannian_gradient_descent}{RGD}{}}
        \State $\newtarget{def:v_k_sup_x}{\vkx[k]} \gets -\exponinv{\xk}(\yk)/\lambda$  \Comment{Approximate subgradient}
        \State $\newtarget{def:iterate_z_x}{\zx[k]} \gets \exponinv{\xk}(\expon{\yk[k-1]}(\zybar[k-1][k-1])) - \etak \vkx[k]$ \Comment{Mirror Descent step} \label{line:MD_step}
        \State $\newtarget{def:iterate_z_y}{\zy} \gets \Gamma{\xk}{\yk}(\zx) + \exponinv{\yk}(\xk)$ \Comment{Moving the dual point to $\Tansp{\yk}\M$}
        \State $\newtarget{def:iterate_z_y_bar}{\zybar} \gets \Pii{\bar{B}(0, \D)}(\zy) \in \Tansp{\yk}\M$ \Comment{Easy projection done so the dual point is not very far} \label{line:dual_projecting_step}
    \ENDFOR
    \State \textbf{return} $\yk[T]$.
\end{algorithmic}
\end{algorithm}

We find the answer in implicit methods. An implicit Euclidean (sub)gradient descent step is one that computes, from a point $\xk \in \newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$, another point $\ykast = \xk - \lambda v_k \in \newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$, where $v_k \in \partial(\f + \indicator{\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}})(\ykast)$, is a subgradient of $\f+\indicator{\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}}$ at $\ykast$. Intuitively, if we could implement a Riemannian version of an implicit \newlink{def:acronym_gradient_descent}{GD}{} step then it should be possible to still compensate the regret of the other algorithm and keep all the iterates in the set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$. Computing such an implicit step is computationally hard in general, but we show that approximating the proximal objective $\newtarget{def:proximal_objective}{\hk}(y) \stackrel{\mathrm{\scriptscriptstyle def}}{=} \f(y) + \frac{1}{2\lambda} \newlink{def:distance}{d}(\xk, y)^2$ with enough accuracy yields an approximate subgradient that can be used to obtain an accelerated algorithm as well.
In particular, we provide an accelerated scheme for which we show that the error incurred by the approximation of the subgradient can be bounded by some terms we can control, cf. \cref{lemma:the_suitable_approximate_prox_has_the_property_needed_in_alg1}, namely a small term that appears in our Lyapunov function and also a term proportional to the squared norm of the approximated subgradient, which only increases the final convergence rates by a constant. We also provide a warm start in \cref{lemma:warm_start_riemannian_criterion_2} and show that an unaccelerated linearly convergent subroutine initialized at the warm-started point achieves the desired accuracy of the subproblem fast, cf. \cref{remark:RGD_can_be_used_as_subroutine}.
This proximal approach works by exploiting the fact that the Riemannian Moreau envelop is g-convex and smooth

in Hadamard manifolds \citep{azagra2005inf} and that the subproblem $\hk$, defined with our $\lambda=\zeta{2\D}/\L $, is strongly g-convex and smooth with a condition number that only depends on the geometry. For this reason, a local algorithm like the one in \citep{criscitiello2022negative} can be implemented in balls whose radius is independent on the condition number of $\f$. Besides these steps, we use a coupling of the approximate implicit \newlink{def:acronym_riemannian_gradient_descent}{RGD}{} and of a mirror descent (\newtarget{def:acronym_mirror_descent}{\newlink{def:acronym_mirror_descent}{MD}{}}) algorithm, along with a technique in \citep{kim2022accelerated} to move dual points to the right tangent spaces without incurring extra geometric penalties, that we adapt to work with dual projections, cf. \cref{lemma:compensated_geometric_penalty}. Importantly, the \newlink{def:acronym_mirror_descent}{MD}{} algorithm keeps the dual point close to the set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ by using the projection in Line \ref{line:dual_projecting_step}, which implies that the point $\xk$ is close to $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ as well, and this is crucial to keep low geometric penalties. This \newlink{def:acronym_mirror_descent}{MD}{} approach is a mix between follow-the-regularized-leader algorithms, that do not project the dual variable, and pure mirror descent algorithms that always project the dual variable. In the analysis, we note that partial projection also works, meaning that defining a new dual point that is closer to all of the points in the feasible set but without being a full projection leads to the same guarantees. Because we use the mirror descent lemma over $\Tansp{\yk}\M$, what we described translates to: we can project the dual $\zy$ onto a ball defined on $\Tansp{\yk}\M$ that contains the pulled-back set $\exponinv{\yk}(\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}})$ and by means of that trick we can keep the iterates $\xk$ close to $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$. And at the same time, the point for which we prove guarantees, namely $\yk$, is always in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$.

Finally, we instantiate our subroutine with the algorithm in \citep{criscitiello2022negative}, in balls of radius independent on the condition number of $\f$ and show in \cref{thm:instantiation} that if we iterate this approximate implementation of a ball optimization oracle, we obtain convergence at a globally accelerated rate. In \cref{sec:other_subroutine} we provide a warm start that allows any Riemannian constrained linearly convergent algorithm to serve as subroutine for our algorithm and we provide another subroutine that allows to reduce our geometric penalties, provided that one can implement the convex projection operator defined in this section. We note \citep[Thm. 15]{zhang2016first} provided a claimed linearly convergent algorithm for constrained strongly g-convex smooth problems, and thus in principle it could be used for our subroutine after the warm start. Unfortunately, we noticed that the proof is flawed when the optimization is constrained. The first inequality in their proof only holds in general for unconstrained problems and not for projected Riemannian gradient descent, not even for the Euclidean constrained case. Thus, to the best of our knowledge there is no convergence analysis for this metric-projected \newlink{def:acronym_riemannian_gradient_descent}{RGD}{} in this setting. In \cref{sec:RGD_and_proj_RGD} we provide an analysis for this algorithm when the diameter of the feasible set is smaller than a constant ($\zetad<2$) and the global minimizer is inside of the set. We show in \cref{prop:metric_projection_in_balls} that metric projections onto Riemannian balls are simple to compute for uniquely geodesic sets, which implies that both our warm start and projected \newlink{def:acronym_riemannian_gradient_descent}{RGD}{} in the aforementioned setting admit a simple implementation. Finally, we include in \cref{thm:rgd} a result on the convergence of unconstrained \newlink{def:acronym_riemannian_gradient_descent}{RGD}{} with curvature independent rates.

We leave the proofs of most of our results to the appendix and state our main theorems below. Using the insights explained above, we show the following inequality on $\psik$, defined below, that will be used as a Lyapunov function to prove the convergence rates of \cref{alg:accelerated_gconvex}.

\begin{proposition}\label{thm:psi_is_lyapunov}\linktoproof{thm:psi_is_lyapunov}
    By using the notation of \cref{alg:accelerated_gconvex}, let
\[
    \newtarget{def:function_for_Lyapunov_analysis}{\psik} \stackrel{\mathrm{\scriptscriptstyle def}}{=} \Ak(\f(\yk)-\f(\xast)) + \frac{1}{2} \norm{\zy-\exponinv{\yk}(\xast)}_{\yk}^2 + \frac{\xi-1}{2} \norm{\zy}_{\yk}^2.
\]
    Then, for all $k\geq 1$, we have $(1-\Deltak)\psik \leq \psik[k-1]$.
\end{proposition}

With this proposition, we can show the convergence of \newlink{def:Riemacon}{Riemacon}{} for g-convex functions.

\begin{theorem}\label{thm:g_convex_acceleration}\linktoproof{thm:g_convex_acceleration}
    Let $\M$ be a finite-dimensional Hadamard manifold of bounded sectional curvature, and consider $f:\NN\subset\M\to\R$ be a g-convex differentiable function in a compact g-convex set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}} \subset\NN$ of diameter $\D$, $\xast \in \argmin_{x\in\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}} \f(x)$, and $\RInit \stackrel{\mathrm{\scriptscriptstyle def}}{=} \newlink{def:distance}{d}(\xInit, \xast)$. For any $\epsilon > 0$, \cref{alg:accelerated_gconvex} yields an $\epsilon$-minimizer $\yk[T] \in \newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ after $T=\bigo{\zetad\sqrt{\frac{\L \RInit[2]}{\epsilon}}}$ iterations. If the function is $\mu$-strongly convex then, via a sequence of restarts, we converge in $\bigo{\zetad\sqrt{\frac{\L}{\mu}}\log(\frac{\mu\RInit[2]}{\epsilon})}$ iterations.
\end{theorem}

We note that a straightforward corollary from our results is that if we can compute the exact Riemannian proximal point operator and we use it as the implicit gradient descent step in Line \ref{line:subroutine} of \cref{alg:accelerated_gconvex}, then the method is an accelerated proximal point method. One such Riemannian algorithm was previously unknown in the literature as well. Note we do not require smoothness of $\f$. Also, we emphasize that our \cref{alg:accelerated_gconvex} only needs to query the gradient of $\f$ at points in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ and the $\L $-smoothness and $\mu$-strong g-convexity of $\f$ only need to hold in $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$. This is relevant because in Riemannian manifolds the condition number $\L /\mu$ in a set can increase with the size of the set, cf. \citep[Proposition 27]{martinez2020global}. Intuitively, although there are twice differentiable functions defined over the Euclidean space whose Hessian is constant everywhere, in other Riemannian cases the metric may preclude having such global condition and the larger the set is the larger the minimum possible condition number becomes. Compare this, for instance, with the bounds on the Hessian's eigenvalues of the squared-distance function in \cref{fact:hessian_of_riemannian_squared_distance}.

Finally, we instantiate \cref{alg:accelerated_gconvex} to implement approximate ball optimization oracles in an accelerated way. We show that applying these oracles sequentially leads to global accelerated convergence. Moreover, we show that the iterates do not get farther than $2\RR$ from $\xastg$, which ultimately leads to the geometric penalty being a function of $\zetar$ and not on the condition number of $\f$. For the subroutine in Line \ref{line:subroutine} of \cref{alg:accelerated_gconvex}, we use the algorithm in \citep[Section 6]{criscitiello2022negative}, and for that we require the following.

\begin{assumption}\label{assump:bounded_curvature_tensor}
    Let $\newtarget{def:curvature_tensor}{\curvtensor}$ be the curvature tensor of a Riemannian manifold $\M$. Its covariant derivative is $\nabla \curvtensor = 0$.
\end{assumption}

Locally symmetric manifolds, like the SPD matrix manifold, manifolds of constant sectional curvature, $\operatorname{SO}(n)$, the Grasmannian manifold, are all manifolds such that $\nabla \curvtensor = 0$. We argue that this assumption is mild, since in particular these manifolds cover all of the applications in \cref{sec:introduction}.

\begin{algorithm}
    \caption{Boosted Riemacon: ball optimization boosting of a \newlink{def:Riemacon}{Riemacon}{} instance (\cref{alg:accelerated_gconvex})}
    \label{alg:instance_of_riemacon}

\begin{algorithmic}[1]
    \REQUIRE Differentiable function $\f:\NN\subset\M\to\R$ that is $\L $-smooth and $\mu$-strongly g-convex in $\ball(\xastg, 3\RR)\subset \NN$; initial point $\xInit \in \NN$; bound $\RR \geq \newlink{def:distance}{d}(\xInit, \xastg)$; accuracy $\epsilon$.

       \newlink{def:Riemacon_SC}{RiemaconSC}{}: The strongly convex version of \cref{alg:accelerated_gconvex} in \cref{thm:g_convex_acceleration} (cf. {\hyperlink{proof:thm:g_convex_acceleration}{its proof}}).
    \vspace{0.1cm}
    \hrule
    \vspace{0.1cm}

    \State \textbf{if} $2\RR \leq (46\RR \abs{\kmin} \zeta{2\RR})^{-1}$ \textbf{then return} $\text{\newlink{def:Riemacon_SC}{RiemaconSC}}(\ball(\xInit, \RR), \xInit, \f, \epsilon)$ \label{line:if_D_equal_R_one_single_Riemacon} \label{line:first_if_in_instanced_alg}
    \State Compute $\D$ such that $\D = (46\RR \abs{\kmin} \zeta{\D})^{-1}$. Alternatively, make $\D\gets (70 \RR\abs{\kmin})^{-1}$.\label{line:computing_D_in_instanced_alg}

    \State $\newtarget{def:total_number_of_iterations_of_instanced_algorithm}{\newlink{def:total_number_of_iterations_of_instanced_algorithm}{T}} \gets \lceil (4\RR/\D) \ln(\L\RR^2/\epsilon) \rceil$; \ \  $\newtarget{def:accuracy_epsilon_prime_in_ball_subproblem}{\newlink{def:accuracy_epsilon_prime_in_ball_subproblem}{\oldepsilon'}} \gets \min\{\D\epsilon/(8\RR),\mu\RR^2/(2\newlink{def:total_number_of_iterations_of_instanced_algorithm}{T}^2)\}$
    \FOR {$k = 1 \textbf{ to } \newlink{def:total_number_of_iterations_of_instanced_algorithm}{T}$}
       \State $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}_k \gets \ball(x_{k-1}, \D/2)$
       \State $x_{k} \gets \text{\newlink{def:Riemacon_SC}{RiemaconSC}}(\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}_k, x_{k-1}, \f, \newlink{def:accuracy_epsilon_prime_in_ball_subproblem}{\oldepsilon'})$ \Comment{\citep{criscitiello2022negative} as subroutine} \label{line:riemacon_w_subroutine}
    \ENDFOR
    \State \textbf{return} $x_{\newlink{def:total_number_of_iterations_of_instanced_algorithm}{T}}$.
\end{algorithmic}
\end{algorithm}

\begin{theorem}\label{thm:instantiation}\linktoproof{thm:instantiation}
    Let $\M$ be a finite-dimensional Hadamard manifold of bounded sectional curvature satisfying \cref{assump:bounded_curvature_tensor}. Consider $f:\NN\subset\M\to\R$ be an $\L$-smooth and $\mu$-strongly g-convex differentiable function in $\ball(\xastg, 3\RR)$, where $\xastg$ is its global minimizer and where $\RR \geq \newlink{def:distance}{d}(\xInit, \xastg)$ for an initial point $\xInit$. For any $\epsilon > 0$, \cref{alg:instance_of_riemacon} yields an $\epsilon$-minimizer after $\bigotilde{\zetar^2\sqrt{\L/\mu}\log(\L\RR^2/\epsilon)}$ calls to the gradient oracle of $\f$. By using regularization, this algorithm $\epsilon$-minimizes the g-convex case ($\mu=0$) after $\bigotilde{\zetar^2\sqrt{\zetar+\L \RR^2/\epsilon}}$ gradient oracle calls.
\end{theorem}

In sum, the algorithm enjoys the same rates as \newlink{def:acronym_accelerated_gradient_descent}{AGD}{} in the Euclidean space up to a factor of $\zetar^2 = \RR^2\kmin[2]\coth^2(\RR\sqrt{\abs{\kmin}}) \leq (1+\RR\cdot\abs{\kmin})^2$ (our geometric penalty) and up to universal constants and $\log$ factors. Note that as the minimum curvature $\kmin$ approaches $0$ we have $\zetar \to 1$. We can reduce the geometric penalties by using a convex projection subroutine, cf. \cref{sec:other_subroutine}.

\section{Our results and related work}\label{sec:related_work}

We compare our results with previous works. We have summarized most of the following discussion in \cref{table:comparisons:riemannian}. We include Nesterov's \newlink{def:acronym_accelerated_gradient_descent}{AGD}{} in the table for comparison purposes\footnote{Note that the original method in \citep{nesterov1983method} needed to query the gradient of the function outside of the feasible set, and this was later improved to only require queries at feasible points \citep{nesterov2005smooth} as in our work, hence our choice of citation in the table.}.
There are some works on Riemannian acceleration that focus on empirical evaluation or that work under strong assumptions \citep{liu2017accelerated, alimisis2019continuous, huang2019extending, alimisis2020practical, lin2020accelerated}, see \citep{martinez2020global} for instance for a discussion on these works. We focus the discussion on the most related work with guarantees. \citep{zhang2018towards} obtain an algorithm that, up to constants, achieves the same rates as \newlink{def:acronym_accelerated_gradient_descent}{AGD}{} in the Euclidean space, for $\L$-smooth and $\mu$-strongly g-convex functions but only \textit{locally}, namely when the initial point starts in a small neighborhood $N$ of the minimizer $\xastg$: a ball of radius $O((\mu/\L)^{3/4})$ around it. \citep{ahn2020nesterov} generalize the previous algorithm and, by using similar ideas as in \citep{zhang2018towards} for estimating a lower bound on $\f$, they adapt the algorithm to work globally, proving that it eventually decreases the objective as fast as \newlink{def:acronym_accelerated_gradient_descent}{AGD}{}. However, as \citep{martinez2020global} noted, it takes as many iterations as the ones needed by Riemannian gradient descent (\newtarget{def:acronym_riemannian_gradient_descent}{\newlink{def:acronym_riemannian_gradient_descent}{RGD}{}}) to reach the neighborhood of the previous algorithm. The latter work also noted that in fact \newlink{def:acronym_riemannian_gradient_descent}{RGD}{} and the algorithm in \citep{zhang2018towards} can be run in parallel and combined to obtain the same convergence rates as in \citep{ahn2020nesterov}, which suggested that for this technique, full acceleration with the rates of \newlink{def:acronym_accelerated_gradient_descent}{AGD}{} only happens over the small neighborhood $N$ in \citep{zhang2018towards}. Note however that \citep{ahn2020nesterov} show that their algorithm will decrease the function value faster than \newlink{def:acronym_riemannian_gradient_descent}{RGD}{}, but this is not quantified.  \citep{jin2021riemannian} developed a different framework, arising from \citep{ahn2020nesterov} but with the same guarantees for accelerated first-order methods. We do not feature it in the table. \citep{criscitiello2022negative} showed, under mild assumptions, that in a ball of center $x\in\M$ and radius $O((\mu/\L)^{1/2})$ containing $\xastg$, the pullback function $\f\circ\expon{x}:\Tansp{x}\M\to\R$ is Euclidean, strongly convex, and smooth with condition number $O(\L/\mu)$, so \newlink{def:acronym_accelerated_gradient_descent}{AGD}{} yields local acceleration as well. In short, acceleration is possible in a small neighborhood because there the manifold is almost Euclidean and the geometric deformations are small in comparison to the curvature of the objective. These techniques fail for the g-convex case since the neighborhood becomes a point ($\mu/\L = 0$).

Finding fully accelerated algorithms that are \textit{global} presents a harder challenge. By a fully accelerated algorithm we mean one with rates with same dependence as \newlink{def:acronym_accelerated_gradient_descent}{AGD}{} on $\L $, $\epsilon$, and if it applies, on $\mu$. \citep{martinez2020global} provided such algorithms for g-convex functions, strongly or not, defined over manifolds of constant sectional curvature and constrained to a ball of radius $\RR$. The convergence rates initially had large constants with respect to $\RR$ but were later improved, cf. \cref{table:comparisons:riemannian}. \citet{kim2022accelerated} designed global algorithms with the same rates as \newlink{def:acronym_accelerated_gradient_descent}{AGD}{} up to universal constants and a factor of $\zetad$, their geometric penalty. However, they need to assume that the iterates of their algorithm remain in their feasible set $\newlink{def:geodesically_convex_feasible_region}{\mathcal{X}}$ and they point out on the necessity of removing such an assumption, which they leave as an open question. Our work solves this question for a wide class of Hadamard manifolds. In their technique, they show they can use the structure of the accelerated scheme to \textit{move} lower bound estimations on $\f(\xastg)$ from one particular tangent space to another without incurring extra errors, when the right Lyapunov function is used. By \textit{moving} lower bounds here we mean finding suitable lower bounds that are simple (a quadratic in their case), when pulled-back to one tangent space, if we start with a similar bound that is simple when pulled-back to another tangent space.

\paragraph{Lower bounds.}
In this paragraph, we omit constants depending on the curvature bounds in the {$\text{big-}O$} notations for simplicity. \citep{hamilton2021no} proved an optimization lower bound showing that acceleration in Riemannian manifolds is harder than in the Euclidean space. \citep{criscitiello2022negative} largely generalized their results. They essentially show that for a large family of Hadamard manifolds, there is a function that is smooth and strongly g-convex in a ball of radius $\RR$ that contains the minimizer $\xastg$, and for which finding a point that is $\RR/5$ close to $\xastg$ requires $\bigomegatilde{\RR}$ calls to the gradient oracle. Note that these results do not preclude the existence of a fully accelerated algorithm with rates $\bigotilde{\RR} +$\newlink{def:acronym_accelerated_gradient_descent}{AGD}{} rates, for instance. A similar hardness statement is provided for smooth and only g-convex functions. Also, reductions as in \citep{martinez2020global} evince this hardness is also present in this case.

\paragraph{Handling constraints to bound geometric penalties.}\hypertarget{sec:handling_constraints_to_bound_geometric_penalties}{} In our algorithm and in all other known fully accelerated algorithms, learning rates depend on the diameter of the feasible set. This is natural: estimation errors due to geometric deformations depend on the diameter via the constants $\zeta{\D}$, $\delta{\D}$, the cosine-law inequalities \cref{lemma:cosine_law_riemannian}, or other analogous inequalities, and the algorithms take these errors into account. All other previous works are not able to deal with any constraints and hence they simply assume that the iterates of their algorithms stay within one such specified set, except for \citep{martinez2020global} and \citep{criscitiello2022negative} that enforce a ball constraint, as we explained above. However, these two works have their applicability limited to spaces of constant curvature and to local optimization, respectively. Note that even if one could show that given a choice of learning rate, convergence implies that the iterates will remain in some compact set, then because the learning rates depend on the diameter of the set, and the diameter of the set would depend on the learning rates, \emph{one cannot conclude from this argument that the assumption these works make is going to be satisfied}. In contrast, in this work, we design a general accelerated framework and an instance of it that keep the iterates bounded in a set we \emph{pre-specify}, effectively bounding geometric penalties while we do not need to resort to any other extra assumptions, solving the open question in \citep{kim2022accelerated}.

Some other works study and use Riemannian metric projections, see \citep{walter1974metric, hosseini2013metric, barani2013metric, bacak2014convex, zhang2016first} and references therein. Among them, \citep{zhang2016first} introduced several deterministic and stochastic first-order methods that use metric-projection oracles.

\paragraph{Riemannian proximal methods.} There are some works that study proximal methods in Riemannian manifolds, but most of them focus on asymptotic results or assume the proximal operator can be computed exactly \citep{wang2015convergence, bento2017iteration, bento2016new, khammahawong2021tseng, chang2021inertial}. The rest of these works study proximal point methods under different inexact versions of the proximal operator as ours and they do not show how to implement their inexact version in applications, like in our case of smooth and g-convex optimization. In contrast, we implement the inexact proximal operator with a first-order method \citep{ahmadi2014convergence} provide a convergence analysis of an inexact proximal point method but when applied to optimization they assume the computation of the proximal operator is exact. \citep{tang2014rate} uses a different inexact condition and proves linear convergence, under a growth condition on $\f$. \citep{wang2016proximal} obtains linear convergence of an inexact proximal point method under a different growth assumption on $\f$ and under an absolute error condition on the proximal function. None of these methods presented acceleration.

\section{Conclusion and future directions}\label{sec:future_work}
In this work, we pursued an approach that, by designing and making use of inexact Riemannian proximal methods, yielded accelerated optimization algorithms. Consequently we were able to work without an undesirable assumption that most previous methods required, whose potential satisfiability is not clear: that the iterates stay in certain specified geodesically-convex set without enforcing them to be in the set. A future direction of research is the study of whether there are algorithms like ours that incur even lower geometric penalties or that do not incur $\log(1/\epsilon)$ factors. Determining whether the convergence rates of fully accelerated algorithms necessarily incur a geometric factor is an interesting open problem: current lower bounds only require an additive geometric penalty and the rate of unaccelerated unconstrained \newlink{def:acronym_riemannian_gradient_descent}{RGD}{} in \citep[Thm. 15]{zhang2016first} does present an additive geometric constant only while all known accelerated methods have a multiplicative geometric constant in their rates. Note that for the local algorithms in \cref{table:comparisons:riemannian}, this factor is a constant. Another interesting direction consists of studying generalizations of our approach to more general manifolds, namely the full Hadamard case, and manifolds of non-negative or even of bounded sectional cuvature.

\clearpage

\acks{
We thank Christophe Roux for proofreading and bringing an error to our attention in an early version of this draft. David Martínez-Rubio was partially funded by the DFG Cluster of Excellence MATH+ (EXC-2046/1, project id 390685689) funded by the Deutsche Forschungsgemeinschaft (DFG).
}

\printbibliography[heading=bibintoc]

\clearpage

redpajama_set_name: RedPajamaArXiv


Audra is a dual British and American citizen originally hailing from California, and raised in England. She started her career in entertainment at just 14 years of age when she was discovered by the prestigeous modeling agency NEXT Model Management.
Audra soon found herself gracing the pages of magazines such as Cosmopolitan and Elle, she became the face of Halston's fragrance 'Unbound', and strutted down countless runways for major designers such as Michael Kors and Alaia. Audra, now a veteran in the modeling industry gained extensive knowledge in the world of fashion and beauty.
Audra has also acted in numerous comedic projects such as guest starring on the sitcom 'Rules of Engagement', Kevin Spacey's 'Father of Invention'. She has also done dozens of commercials most notably, Bud Light's 2013 Super Bowl commercial directed by Samuel Bayer.
Audra was lucky enough to start her hosting career working with legend, Larry King, for his online network, Ora Tv. She's a freelance host and writer for Ora Tv's NewsBreakers where she covers news, trends and what's buzzing in social media. Her humor and unfiltered wit landed her the job of additionally hosting Ora's satirical news series called WTFark. Audra also landed another show with Ora where she covers fun facts about everything memorabilla called The Collectors Cafe. You can also find Audra hosting a variety show that airs in the UK on Sky called NubTv and is currently hosting a news show for Fashion Television.
When Audra isn't in front of the camera, you can find her behind it practicing her sharp shooting skills, discovering new and cool spots on her travels, secretly playing video games, earning her black belt shopping skills and watching soccer and American football with the boys!


A recent article by Presbyterian Pastor Tim Keller calling for an openness to both American political parties by American Christians has gained a lot of attention. Writing in the New York Times, Pastor Keller argues that the legacy of the Christian Church's position on social issues does not fit into our contemporary political situation today.
I think Pastor Keller is spot on in his analysis here. Many of our ongoing political debates surround issues which are of vital importance to our day-to-day lives. He is also stressing that many times people who are considering the Christian worldview are more likely to see Christians' political views before they actually see Christianity. To this I am also sympathetic: in the Christian worldview salvation and redemption are not dependent on who is in a place of public office but instead on what Jesus Himself did.
However, while I do agree with some of his points, I think Pastor Keller makes a dangerous mistake in assuming moral equivalency between both of the major political parties in the United States today, and the positions they take on certain important issues; most importantly, the issue of abortion. As a member of the Presbyterian Church of America myself, I am deeply concerned about the lack of discretion by some leaders in the denomination on issues surrounding justice and human rights, particularly in supporting the preborn.
Noticeably absent in Keller's piece is any real reference to abortion or where the American Left and Right generally fall on the issue. The only reference to anything remotely related to the issue of abortion is the legitimate basis for sexual interaction, which is in the context of marriage.
The question has to be asked: why leave out the most heated debate between the parties for the last half-century? Why leave out the issue of abortion?
Whether Pastor Keller merely neglected to mention abortion or whether he left it out intentionally, obviously, as Greg Koukl famously stated, if the preborn are not human, abortion is no big deal morally (and therefore, politically), but if the preborn are in fact human beings, then abortion needs to be taken into extremely serious consideration by anyone looking into how the Christian worldview impacts political activism.
As Keller states regarding the issue of race relations, racism is a sin because it violates the command to "love your neighbor." However, if abortion intentionally kills an innocent human being, it too violates the command to "love one's neighbor." Therefore, abortion is also a sin that needs to be opposed by committed Christians just as racism.
This is why Keller's assumption of moral equivalency is so dangerous. By finding ways to compromise the philosophies of both political parties which, admittedly, can be absurdly complex at times, Keller seems to be acting as if the preborn are not fully human.
Which modern American political party has given the most support to abortion in the last few years? The answer is clear to anyone paying attention.
As Ramesh Ponnuru documents in his book The Party of Death, while there are dissenters among both Republicans and Democrats, the support for abortion has been represented most strongly among the Democrat Party in recent years. One only need look at the way much of the American Left has come to the defense of abortion giant Planned Parenthood since the undercover investigation and subsequent revelation by the Center For Medical Progress that the organization was trafficking the bodily remains of babies killed in abortion.
Not only that, but the opposition to pro-life legislation at both the state and federal levels of government from the American Left has been well-documented. Consider former-President Bill Clinton's opposition to the proposed ban on Partial-Birth Abortion. Clinton is not the only one; other Democratic political candidates have endorsed the procedure as well (also see: Natural Rights and the Right to Choose by law professor Hadley Arkes, who helped write the bill to ban the procedure, for a detailed overview of the debate in the US Congress).
There is also a case of Democrats, including then-Senator Obama, voting against and standing in opposition to bills like the Born-alive Infant Protection Act, to protect babies who are inadvertently born prematurely during botched abortions.
As Hadley Arkes indicates, the proposed law raised the question, "What does the right to an abortion entail?" And as the votes against the born-alive law go to show, the mindset is that a legal right to abortion implies the right to a dead baby (Natural Rights and the Right to Choose). The list could go on, up to and including the disproportionate rate at which minority children are killed in the womb, the methods of abortion itself, and very recent bills in California forcing pro-life pregnancy centers to offer referrals to abortion clinics.
Then I have a question for Pastor Keller: are the preborn examples of an oppressed neighbor that we must care for? If so, parties and groups that openly support the killing of our neighbors must be opposed.
We do have a good reason to reject the proposal that Christianity can easily fit with the key philosophies of either political party. This is not to claim that those on the American Left are abhorrently evil or worthy of condemnation or ridicule (they are human beings with intrinsic value as well as fellow neighbors and Americans). But it is possible to still be mistaken about the truth, even while having noble intentions.
The same mistake is being made by assuming a party which endorses and defends the intentional killing of innocent human beings is an acceptable party for Christians to endorse. The victims of oppression may have changed, but our moral obligations to protect the innocent have not.
If one party has endorsed intentionally harming the innocent, they must be opposed. If an entire political party were openly in support of slavery and vowed to back candidates who would use their office to defend the practice legislatively, I am sure Pastor Keller would say Christians had a moral duty to oppose that party until they changed their position on the issue. And if another party has offered to protect the innocent, then they need to be supported.
If an entire political party was opposing the unjust treatment of an entire class of human beings, would Keller say Christians should not offer support to the party striving to see justice accomplished for that group? Would he then say Christianity could not fit into the two-party system of the time? Clearly, one party is more committed to the protection of our neighbors than its opponent.
To be clear, there have been dissenters on both sides of the political aisle, with well-known Republicans having supported abortion and a small but dedicated group of Democrats working to reverse their party's wholesale support of abortion-on-demand. Unfortunately, the influence of pro-life Democrats has been and continues to be small and dissenters have had little, if any effect in reversing the positions of either party. Until the Democratic Party has effectively reversed course, Christians committed to justice need to think clearly about what accomplishing justice for the innocent looks like.
Lastly, while Keller is correct that Christians should care about issues like racial hostility and poverty, no one supports intentionally kill our poor or non-white neighbors. On the other hand, there are entire institutions and a major political party dedicating themselves to killing our neighbors before they are even born. It is easy to assume the two major political parties are equivalent in upholding the values of a Christian worldview if one has simply not taken the time to consider what is actually being supported.
The views and opinions expressed in this article are those of the author and do not necessarily reflect the official position of FreeThinking Ministries. This article was originally published by Human Defense and has been re-published with the author's permission. FreeThinking Ministries would be honored to publish a response from Pastor Keller if he is so inclined.
Nathan is a staff apologist for the Life Training Institute, equipping pro-life advocates to make the case for life. Also a contributing writer at The Millenial Review and CampusReform.


Filing for bankruptcy really isn't the end of the world, but it does have heavy consequences that will affect your finances in the coming years. I've discovered that in many cases, focusing efforts on developing a bright future is the best way for individuals to tackle their bankruptcy and subsequent recovery. To do this, however, individuals have to grasp exactly what bankruptcy entails so they can properly budget, plan, and rebuild their wealth in the most efficient way possible.
One of the most frequent questions I get asked pertains to how bankruptcy will affect child support payments. Whilst this topic may seem relatively straightforward, I've found that it causes a lot of misunderstanding so today we're going to take a closer look and attempt to resolve some of that confusion.
Does bankruptcy cover child support debts?
Even though bankruptcy releases you from a wide range of debts, child support is not one of them. If you owe a large amount of money in child support when you file for bankruptcy, it will not be released in bankruptcy so it's best to speak with the Department of Human Services (DHS) and negotiate a repayment plan. If, for whatever reason, you feel the assessment given by the DHS is incorrect, you can contest this.
The DHS is in charge of regulating and working with separated parents on child support assessments. To determine how much child support you must pay, the DHS examine both your income and your care percentage of the children involved. By utilising your latest tax return as a benchmark, the DHS will use these numbers to ascertain your anticipated income for the upcoming year. This showcases the value of keeping your tax returns up to date, and any adjustments to your circumstances should be reported to the DHS immediately.
The DHS define a dependent as someone who lives with you most of the time and earns under $3,539 each year.
As a result, every 50 cents you earn over your income threshold will be used to pay the debts in your bankrupt estate.
For instance, if you earn $110,000 annually before tax, you'll likely be paying close to $30,500 each year in tax. Your assessable income would therefore be roughly $79,500. Assuming you have no other income and no dependents live with you at home, your trustee would calculate your bankruptcy payments as follows:.
($79,500 – $55,837.60) ÷ 2 = $11,831.20 (or roughly $986 monthly).
Your child support contributions are subtracted from your taxable income so the more child support you pay, the less money gets contributed to your bankruptcy estate. Using the previous example, if you are required to pay $15,000 in child support payments every year, your assessable income would be reduced from $79,500 (income after tax) to $64,500.
After providing your trustee with a copy of your child support assessment from the DHS, your trustee would determine your bankruptcy payments as follows:.


The Album from the South Africans, Biorhythm is available now!
Biorhythm consists of Carl Sharples and Kerion Grieve. These artists have been some of the most exciting producers who have moved their production skills onto dance floors, Born and bred in Cape Town. Bringing powerful rhythms to major events in South Africa and abroad, they have introduced their new cutting edge psychedelic sound to the world of trance. They have got their advanced diploma in sound engineering, are the founders of The Village and the village records, they have been making music for over 9 years and have a variety of aliases, where they express their musical talents in other genre's. You can buy this album by clicking on image of the release Keep it Psy!


The enemies of the Islamic Revolution are the same people who entered the region 10, 15 years ago with the slogan of providing security. Take a look today and see where in the region security exists! Insecurity has engulfed the entire region. West Asia and North Africa are filled with insecurity.
When they attacked Afghanistan, their slogan was fighting against terrorism, but today, terrorism has engulfed the entire region- and what terrorism it is! It is a kind of savage and violent terrorism, one in which individuals burn people – their enemies – alive and in front of everyone. And then they use different technical means to display this directly to the people throughout the world. It is this kind of terrorism. Today, takfiri elements are like this. They kill a child in front of its mother and they kill parents in front of their children. They had come to eliminate terrorism from the region with this slogan and claim. I am not saying that this was their real goal, rather this was their slogan- but where in the region does terrorism not exist in the present time?
They had come – as they claimed – to establish democracy. Today, the most reactionary, dictatorial and tyrannical regimes are standing upright and continuing their crimes with the help of America and its allies. This is really one of the major problems of America. Today, this problem has entangled American politicians as well. They have really gotten stuck in it. They are supporting such regimes. Over the course of many years, their anti-dictatorship slogans and claims to support human rights have been repudiated because of the existence of such regimes. In the present time, this issue has posed serious questions among American intellectuals, political personalities and thinkers and they have no answers for them.
This enemy is such a creature! This is how the issue of human rights, democracy, terrorism, security and peace has worked out for the enemy who is in front of us. They used to say that they were fighting for peace, but what peace? They have polluted the whole region with war. Where in the region does war not exist? This is the enemy. He whom the Revolution is confronting and he whom you are standing up against is this! It is this creature with all these contradictions, all these weak points and all these ideological and practical shortcomings. One of the characteristics of guarding the Revolution is to open our eyes and to see these things. We should know the enemy. We should both know the Revolution and the enemy. This was related to guarding the Revolution. Of course, if someone wants to speak more about guarding, its meaning and its depth, they can write volumes of books.


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Melipotis jucunda, the merry melipotis moth, is a species of moth in the family Erebidae. It is found in Mexico (Yucatán, Mérida), Colombia, most of the United States, western Canada and northeast Brazil.

The wingspan is 35–42 mm. The ground colour of the forewings is medium grey, with fine longitudinal streaks and black dashes between the veins on the distal portion. The hindwings are pure white to pale grey with dark veins, a thin discal spot and a wide marginal band. There are two or three generations in New Jersey and multiple generations in the southern part of the range.

The larvae feed on Salix bonplandia and Salix wrighti, Acacia species and Calliandra eriophylla.

Subspecies
Melipotis jucunda jucunda
Melipotis jucunda hadeniformis (Behr, 1870) (California)

References

Moths described in 1818
Melipotis

redpajama_set_name: RedPajamaWikipedia


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Chapter 139: Supreme Commander's male beauty trap Original and most updated translations are from volare. If read elsewhere, this chapter has been stolen. Please stop supporting theft.
"......." said Lou Zigui. Big Boss Black swished his tail in agitation before running for the window and escaping outside. A cat like him couldn't stand to listen to that kind of conversation anymore.
How am I supposed to follow up to that?
Lou Zigui was indeed left speechless.
"Hahaha," Ning Xiaoyao said, amused. "I just realized that I have the potential to be a tyrant. That there, 'beloved consort, you are just too beautiful. This emperor will never go to the morning court sessions again,' hahaha~" She burst into laughter.
(#‵′), went Supreme Commander Lou. What kind of balderdash is she spouting?!
Elder Li was unconvinced. Lou Zigui seemed like a scrupulous and methodical man, yet His Majesty could be so happy just by speaking to him? "But Xie Wenyuan isn't dead yet," Elder Li muttered quietly to himself.
"........" said Shadowbolt. Are you saying His Majesty can't laugh until Grand Preceptor Xie dies? What kind of twisted logic is that?
"Announce my arrival," Elder Li pressed Shadowbolt, who was secretly ridiculing the man in his heart. He's only remembering to have me announce his arrival now. You brought so many people with you to Supreme Splendor Hall, so why didn't you remember that then?
"Don't you know how to announce my arrival?" Elder Li asked when he saw that Shadowbolt hadn't budged.
"Afraid of him?" Lou Zigui asked.
Lou Zigui didn't understand. Why would an emperor be afraid of a subject with no military power, who only had stern words on his side? She wasn't even afraid of Xie Wenyuan, but she had a healthy fear of Elder Li.
"But Xie Wenyuan's nothing but a villainous old man!" Ning Xiaoyao said staunchly.
"Have Elder Li and the rest come in," Lou Zigui stood up. They should stop this topic for now. In any case, he could understand Elder Li's words even when he didn't understand Miss Ning most of the time. After Elder Li and his men were summoned into the room and had paid their respects, he asked about Ning Xiaoyao's injury first.
Elder Li knitted his brows and gave Ning Xiaoyao a stern look. "Doctors shouldn't treat themselves. It's highly important for Your Majesty to take care of your dragon body. As the saying goes, a body worth a thousand pieces of gold…" Once again, Elder Li attempted to instruct Ning Xiaoyao with wisdom from sages of the past.
Ning Xiaoyao's forehead broke out in cold sweat as she shot Lou Zigui a pleading look. What in the heck is that old gramps talking about now?
Lou Zigui had no choice but to steer the topic away. "What has Elder come to find His Majesty for?" His question earned an unhappy glare from Elder Li, who'd been interrupted in the middle of his tirade.
"........." said Elder Li and the rest. After something so catastrophic, do you think just filling in the hole will fix everything?
"This subject isn't sleepy, either!" Elder Li was now shouting.
Ning Xiaoyao began to dig her nails into her palms again. She'd purposely chosen safe topics like eating and sleeping, but it'd still earned that old gramp's ire. Is it impossible for us to coexist peacefully? "Old gramps, just what's wrong with you?" Ning Xiaoyao felt a headache coming on.
"Your Majesty is still asking this subject?" Elder Li replied.
"This subject isn't angry!" Elder Li roared.
Save me! Ning Xiaoyao clutched her head. Can't we just talk like sensible people? All of the subjects who'd came with Elder Li backed up a step. They'd never expected to hear such a pointless conversation after entering this room, much less see Elder Li so mad or His Majesty so ready to die.
Everyone now fixated their stares on Lou Zigui. Supreme Commander, why does it sound like you're saying His Excellency Elder's previous words were incomprehensible to His Majesty?
"His Majesty is tired after today's events," Lou Zigui added after realizing his previous statement was a little fishy. Ning Xiaoyao immediately affected looking weak, all the best to make Elder Li go home faster. Elder Li couldn't help but to be amused and chuckled a few times despite his anger. Ning Xiaoyao immediately straightened her posture in her chair, having never seen an old man laugh so unkindly.
"How is Your Majesty planning to deal with Xie Wenyuan?" Elder Li asked Ning Xiaoyao.
This was an irrefutable truth, even to Elder Li.
"Then, what about the refugees who broke into the palace? What will Your Majesty do with them?" another official asked.
"Let them go?" many officials chorused at the reply. Those refugees were conspiring against the state, and we're just letting them go?!
"........." said everyone else. His Majesty was probably the first person to ever curse Elder Li as disgraceful.
Elder Li stared woodenly at Ning Xiaoyao. Ning Xiaoyao had plenty of other comforting things to say, but she lost her courage to say them in the face of his stare.
"I listen to His Majesty," Lou Zigui answered.
"Your Majesty is certainly wise and brilliant," Elder Li then bowed towards Ning Xiaoyao. Recently, His Majesty had taken care of all the refugees. But they'd still believed in hearsay and charged the palace to revolt. Those refugees were nothing more than ungrateful types who'd bit the hand that fed them, and people who thoroughly deserved death. Yet the current emperor was still willing to forgive them. Such breadth of mind made Elder Li himself feel inferior.
Ning Xiaoyao was so shocked that she nearly fell out of her chair. What did I end up saying this time?!
"The sovereign possesses a bosom as vast as the four seas," Elder Li said, still bent over in a bow.
When Lou Zigui saw Ning Xiaoyao blatantly examining her chest, he knew immediately what she was thinking about. His face inexplicably reddened before he patted her on the shoulder so she could stop staring and raise her head. It's not like they'll grow just from your staring.
Elder Li said, "Your Majesty is wise and brilliant, moving this subject to tears of gratitude." His eyes really had turned red. He had waited two successive dynasties for a worthy sovereign. Old Elder Li had it rough, too!
Ning Xiaoyao silently chuckled to herself. How did I end up wise and brilliant again? She couldn't figure it out, but she felt like Elder Li was a very profound and subtle person.
"Then how is Your Majesty planning to deal with the people who raised blades against you?" another official asked.
Lou Zigui rested his hand on Ning Xiaoyao's shoulder and squeezed it slightly.
"Anyone who raised arms can be directed to Supreme Commander to deal with," Ning Xiaoyao answered immediately.
"This subject accepts the decree," Lou Zigui said with a bow.
"Anything else?" Ning Xiaoyao asked Elder Li and the rest.
"That doppelganger?" Elder Li asked.
"Your Majesty is going to search for his family?" Elder Li said uncertainly.


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'Masterplan ignores green gap'
Shinfield residents have criticised the way a blueprint for 2,500 homes on their doorstep has been produced.
Shinfield Parish Council says it objects to the way Wokingham Borough Council has gone about drawing up the masterplan for homes to the south of the M4.
The council, which is campaigning on behalf of residents in the parish, says the masterplan differs to what the council set out in the Core Strategy, which proposes four sites for large-scale housing development across the borough.
Under proposals in the masterplan for land south of the M4, homes would be mainly built to the south west of Shinfield and to the north east of Spencers Wood.
Jacqui Barnes, clerk to the council, said: "This totally ignores the relationship between the current built areas and open countryside."
A consultation on the four masterplans, including Arborfield Garrison, land south of the M4 and sites to the north and south of Wokingham town, ended in March. The council's executive is due to discuss the final masterplans at the end of May.
Wokingham Borough Council

redpajama_set_name: RedPajamaCommonCrawl


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Louis Joseph Freeh (, 6. siječnja 1950.) američki je odvjetnik, poduzetnik i političar, poznat kao bivši direktor FBI-ja u doba administracije predsjednika Clintona i Georgea W. Busha. Član je Republikanske stranke.

Freeh je odrastao u katoličkoj obitelji, a godine 1971. diplomirao je na Sveučilištu Rutgers. Godine 1975. priključio se FBI-ju i služio kao agent u njujorškom uredu. Godine 1981. postao je savezni tužitelj, a 1991. i savezni sudac. Bill Clinton ga je 1993. imenovao, a Senat potvrdio na mjesto direktora FBI-ja.

Na početku mandata FBI je bio pod žestokim kritkama američke javnosti zbog krvavih incidenata u Ruby Ridgeu i Wacou. Freeh je kao tehnokrat pokušao popraviti taj dojam, iako su se kritike nastavile, prvenstevno zbog neuspjeha da se spriječi bombaški napad u Oklahoma Cityju.

Zbog brojnih kritika Freeh je u lipnju 2001. podnio ostavku. Kasnije je u javnosti branio FBI od optužbi za nesposobnost koje su se pojavile nakon Napada 11. rujna 2001. godine. Danas radi kao odvjetnik.

Freeh je 2005. objavio knjigu memoara u kojoj se prilično kritički odnosi prema predsjedniku Clintonu.

Izvori

 Busheva administracija


The 8th Annual Shaadi Bazaar Wedding Expo & Shopping Event was a wonderful event for the brides and shoppers that attended. We were excited to welcome over 25 amazing and talented vendors from the New England area who are ready to help you plan your dream wedding!
We had the talented decor experts Alankar and Love Decor, who will be displayed their stunning mandap and staging designs, caterers Chinese Mirch and Masala Art provided delicious food available for free tastings for brides and grooms and Ula's Snack Shop is bringing some scrumptious snacks for you to try as well!
We're also happy to host a wide variety of photographers and videographers including Art and Memories, Purple Ganesh, BM photography, Epic Film and Newbury Studios and there were a couple of new venues for everyone to check out: Lake Pearl, Best Western Royal Plaza and Trade Center, Hilton Woburn and our host Marriott Burlington. Looking for someone to help plan your big day? Visit the expert wedding planners from Blueprint Occasions!
If you're looking for a DJ and Lighting team, look for the talented teams from Silk Events and Xtreme Productions and check out the large letter lighting designs from Doris Loves USA. Don't forget to make time to get a free henna trial from Henna Cafe and Artistic Henna or get your eyebrows threaded at the Heritage Threading Salon booth and get expert hair and makeup advice from Makeup by Marium!
And of course, we always provide the community with a wide variety of clothing and jewelry vendors so they can dress to impress at every wedding event, Garba, Diwali festivities, etc. Shoppers came in droves to check out fashions from Choonri Collection and Bandhni 4 U, custom bridal party wear from Pleats and Petals and beautiful jewelry from Shikha's Glory. And this year, we' were proud to feature new designer Shy Walk, who debuted her creations at our fashion show!


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The lies are hurting them lol
No Comments on The lies are hurting them lol
THE REUNION OF BLACK FAMILY WORLD WIDE está con Dr.Chukwuma Christopher Osaji y 13 personas más.
25 de marzo de 2013 a las 17:42 ·
All Non-Africans Part Neanderthal, Genetics Confirm.
If your heritage is non-African, you are part Neanderthal, according to a new study in the July issue of Molecular Biology and Evolution. Discovery News has been reporting on human/Neanderthal interbreeding for some time now, so this latest research confirms earlier findings.
Damian Labuda of the University of Montreal's Department of Pediatrics and the CHU Sainte-Justine Research Center conducted the study with his colleagues. They determined some of the human X chromosome originates from Neanderthals, but only in people of non-African heritage.
"This confirms recent findings suggesting that the two populations interbred," Labuda was quoted as saying in a press release. His team believes most, if not all, of the interbreeding took place in the Middle East, while modern humans were migrating out of Africa and spreading to other regions.
The ancestors of Neanderthals left Africa about 400,000 to 800,000 years ago. They evolved over the millennia mostly in what are now France, Spain, Germany and Russia. They went extinct, or were simply absorbed into the modern human population, about 30,000 years ago.
Neanderthals possessed the gene for language and had sophisticated music, art and tool craftsmanship skills, so they must have not been all that unattractive to modern humans at the time.
"In addition, because our methods were totally independent of Neanderthal material, we can also conclude that previous results were not influenced by contaminating artifacts," Labuda said.
This work goes back to nearly a decade ago, when Labuda and his colleagues identified a piece of DNA, called a haplotype, in the human X chromosome that seemed different. They questioned its origins.
Fast forward to 2010, when the Neanderthal genome was sequenced. The researchers could then compare the haplotype to the Neanderthal genome as well as to the DNA of existing humans. The scientists found that the sequence was present in people across all continents, except for sub-Saharan Africa, and including Australia.
"There is little doubt that this haplotype is present because of mating with our ancestors and Neanderthals," said Nick Patterson of the Broad Institute of MIT and Harvard University. Patterson did not participate in the latest research. He added, "This is a very nice result, and further analysis may help determine more details."
David Reich, a Harvard Medical School geneticist, added, "Dr. Labuda and his colleagues were the first to identify a genetic variation in non-Africans that was likely to have come from an archaic population. This was done entirely without the Neanderthal genome sequence, but in light of the Neanderthal sequence, it is now clear that they were absolutely right!"
The modern human/Neanderthal combo likely benefitted our species, enabling it to survive in harsh, cold regions that Neanderthals previously had adapted to.
"Variability is very important for long-term survival of a species," Labuda concluded. "Every addition to the genome can be enriching."
http://news.discovery.com/h…/genetics-neanderthal-110718.htm
← the Banks → Nuclear attacks on America


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Q: How would you determine the transformation matrix? Suppose there exists a linear transformation $T$ where $T: \mathbb{R^3} \to \mathbb{R^5}$ and $T(\textbf{x}) = \text{A} \textbf{x}$. Given
$$ \text{A} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1\\ 2\\ 1 \\3 \end{pmatrix} ,
\quad
\text{A} \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} \hspace{6pt}1 \\ \hspace{6pt}2\\ \hspace{6pt}2\\ \hspace{6pt}2 \\-1 \end{pmatrix} ,
\quad
\text{A} \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 3\\ 4\\ 3 \\2 \end{pmatrix} ,
$$
how can we then determine the transformation matrix $\text{A}$? What are some "standard methods" to do it? Isn't this the same as finding the change-of-coordinates matrix when making a change of basis?

A: Let $v_1=(1,1,1)^T, v_2=(1,2,3)^T$ and $v_3=(0,1,1)^T$ then we verify easily that $B=(v_1,v_2,v_3)$ is a basis of $\Bbb R^3$. Let $B_c$ the canonical basis and $P$ the change matrix from $B$ to $B_c$. Now if $\textbf{x}$ is a vector of $\Bbb R^3$ such that $X=(x,y,z)^T$ is its coordinates  in $B_c$ then $PX=(\alpha,\beta,\gamma)^T$ is its coordinates in $B$ and then
$$T(\textbf{x})=A(PX)=\alpha A v_1+\beta A v_2+\gamma A v_3$$

A:
Isn't this the same as finding the change-of-coordinates matrix when
  making a change of basis?

Totally.
Let $P = \begin{pmatrix} 1 & 1& 0 \\ 1 &2 & 1 \\ 1 & 3& 1\end{pmatrix}$
It's easy to see that $P\in GL_3(\mathbb R)$.
Your equalities can be rewrite:
$$AP= \begin{pmatrix} 1&1&2 \\ 1&2&3\\ 2&2&4\\ 1&2&3 \\3 &-1&2\end{pmatrix}$$
Finally:
$$ A =  \begin{pmatrix} 1&1&2 \\ 1&2&3\\ 2&2&4\\ 1&2&3 \\3 &-1&2\end{pmatrix} P^{-1}$$

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Home » Features » Features » Taking The Road Less Traveled Together
Taking The Road Less Traveled Together
By Mike Viola
Fr. Ron Carignan, OMI stands next to Fr. Leo Dummer, OMI
At every meal, Fr. Ron Carignan, O.M.I. sits at the same table at the Oblate Madonna Residence in San Antonio, Texas. To his right always sits Fr. Leo Dummer, O.M.I. It's an appropriate seating arrangement because Fr. Leo has been Fr. Ron's right-hand-man for more than 50 years.
"God put us together at the start of our ministries," said Fr. Leo. "We've been like a hand and a glove ever since."
Father Ron and Fr. Leo started out as an Oblate Odd Couple. Father Ron is from the Lowell, Massachusetts area outside of Boston. As a child he attended St. Joan of Arc grade school with hundreds of other children. Father Leo grew up on a farm in New Ulm, Minnesota. He attended a one room grade school.
As young men they both felt a calling to the priesthood and joined the Oblate formation program. But throughout their time as seminarians, they never met each other, Fr. Ron studying for the priesthood on the East Coast and Fr. Leo in the Midwest.
In 1959 both men were ordained. Most young Oblates at that time were looking at the foreign missions, and that was causing a shortage of Oblates in domestic ministries. So Fr. Ron and Fr. Leo decided to stay home and go to the mission territory of California. That's where their paths finally crossed.
For about the next 20 years, Fr. Ron and Fr. Leo taught and were administrators at high schools. "We ran top notch schools," said Fr. Ron. "We taught everything and when we left a school we could say it was better than we arrived," said Fr. Leo as he finished his friends thought.
Father Ron had a gift for administrative work and would often become the principal at the school. He would usually appoint Fr. Leo as the vice-principal.
Eventually Fr. Ron took on administrative roles with the Oblates, including being named the Provincial for the Western Province. When he needed a Treasurer, it was no surprise the first, and only person he asked to take the job was Fr. Leo.
"He called and asked me to think about becoming the Treasurer," said Fr. Leo. "Why would I need to think about it? I'll go wherever he asks me to go."
That loyalty took on special meaning in 1997 when both men decided they were not ready for retirement. Father Ron, who was finishing an assignment working at the Oblate General House in Rome, accepted the challenge of becoming the Superior of the Oblates' ministries in Zambia. Father Ron called Fr. Leo to tell him about his decision. Within a short time, Fr. Leo was on a plane headed for Africa.
"I always asked Fr. Leo to take care of the money and that is what he did for the five years we were in Zambia together," said Fr. Ron. "He had every penny accounted for and that really helped us grow the mission there."
One of their main accomplishments in Zambia was the creation of Radio Liseli, a radio station that broadcasts a variety of religious programming throughout the western part of the country. In addition to religious programs, Radio Liseli also broadcasts programs that respond to local community concerns, creating space for people to air their concerns and get them addressed.
"Our work in Zambia required us to chart a daring path, a journey in which we could not be faint of heart or let fear hold us back," said Fr. Ron. "We had to be willing to take the difficult road, and in doing so we discovered hat it was a wonderful place to be."
Today, Fr. Ron and Fr. Leo are again at a place that they consider wonderful. They live at the Oblate Madonna Residence along with about two dozen other retired Oblates. Father Leo has trouble walking so Fr. Ron is there to push him in his wheelchair. Fr. Ron occasionally has trouble remembering things, so Fr. Leo is there to help fill in the gaps.
They both enjoy reading, and one of their favorite books is The Road Less Traveled. The book looks at how to overcome difficulties and challenges in a person's life.
For Fr. Ron and Fr. Leo, they chose to take the road less traveled. And because they took that road together, it was a road filled with much satisfaction, hope and joy.


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Zoocid je pesticid používaný k ochraně rostlin před živočichy, kteří mohou působit ekonomicky neúnosná poškození rostlin. Použití zoocidů v integrované ochraně rostlin je součástí chemických metod, které v současné době zejména v rozvinutých zemích převažují nad ostatními, vzhledem k jednoduchosti použití a finanční nenáročnosti oproti jiným metodám. Dlouhodobé a opakované používání však sebou přináší toxicitu, vznik rezistence a zamoření půdy či vod. Toxicita většinou přichází s nadměrným použitím přípravku. Udává se buď pro člověka nebo pro včely. Rezistence organizmů pak vzniká při opakovaném nasazení stejného přípravku. Zoocidy se podle typů škůdců na které působí dělí na akaricidy (proti roztočům), insekticidy (proti hmyzu), nematocidy (proti háďátkům), moluskocidy (proti slimákům) a rhodenticidy (proti hlodavcům).

Pesticidy
Ochrana rostlin