We introduce a general class of transport distances ${\rm WB}_{\Lambda}$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures distance. Such a distance is defined via a generalized Benamou-Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimization problem. Some recently proposed models, including the Kantorovich-Bures distance and the Wasserstein-Fisher-Rao distance, can naturally fit into ours. We give a complete characterization of the minimizer and explore the topological and geometrical properties of the space $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$. In particular, we show that $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$ is a complete geodesic space and exhibits a conic structure.
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This paper focuses on representing the $L^{\infty}$-norm of finite-dimensional linear time-invariant systems with parameter-dependent coefficients. Previous studies tackled the problem in a non-parametric scenario by simplifying it to finding the maximum $y$-projection of real solutions $(x, y)$ of a system of the form $\Sigma=\{P=0, \, \partial P/\partial x=0\}$, where $P \in \Z[x, y]$. To solve this problem, standard computer algebra methods were employed and analyzed \cite{bouzidi2021computation}. In this paper, we extend our approach to address the parametric case. We aim to represent the "maximal" $y$-projection of real solutions of $\Sigma$ as a function of the given parameters. %a set of parameters $\alpha$. To accomplish this, we utilize cylindrical algebraic decomposition. This method allows us to determine the desired value as a function of the parameters within specific regions of parameter space.
This paper focuses on representing the $L^{\infty}$-norm of finite-dimensional linear time-invariant systems with parameter-dependent coefficients. Previous studies tackled the problem in a non-parametric scenario by simplifying it to finding the maximum $y$-projection of real solutions $(x, y)$ of a system of the form $\Sigma=\{P=0, \, \partial P/\partial x=0\}$, where $P \in \Z[x, y]$. To solve this problem, standard computer algebra methods were employed and analyzed \cite{bouzidi2021computation}. In this paper, we extend our approach to address the parametric case. We aim to represent the "maximal" $y$-projection of real solutions of $\Sigma$ as a function of the given parameters. %a set of parameters $\alpha$. To accomplish this, we utilize cylindrical algebraic decomposition. This method allows us to determine the desired value as a function of the parameters within specific regions of parameter space.
Modelling noisy data in a network context remains an unavoidable obstacle; fortunately, random matrix theory may comprehensively describe network environments effectively. Thus it necessitates the probabilistic characterisation of these networks (and accompanying noisy data) using matrix variate models. Denoising network data using a Bayes approach is not common in surveyed literature. This paper adopts the Bayesian viewpoint and introduces a new matrix variate t-model in a prior sense by relying on the matrix variate gamma distribution for the noise process, following the Gaussian graphical network for the cases when the normality assumption is violated. From a statistical learning viewpoint, such a theoretical consideration indubitably benefits the real-world comprehension of structures causing noisy data with network-based attributes as part of machine learning in data science. A full structural learning procedure is provided for calculating and approximating the resulting posterior of interest to assess the considered model's network centrality measures. Experiments with synthetic and real-world stock price data are performed not only to validate the proposed algorithm's capabilities but also to show that this model has wider flexibility than originally implied in Billio et al. (2021).
Given an $m\times n$ binary matrix $M$ with $|M|=p\cdot mn$ (where $|M|$ denotes the number of 1 entries), define the discrepancy of $M$ as $\mbox{disc}(M)=\displaystyle\max_{X\subset [m], Y\subset [n]}\big||M[X\times Y]|-p|X|\cdot |Y|\big|$. Using semidefinite programming and spectral techniques, we prove that if $\mbox{rank}(M)\leq r$ and $p\leq 1/2$, then $$\mbox{disc}(M)\geq \Omega(mn)\cdot \min\left\{p,\frac{p^{1/2}}{\sqrt{r}}\right\}.$$ We use this result to obtain a modest improvement of Lovett's best known upper bound on the log-rank conjecture. We prove that any $m\times n$ binary matrix $M$ of rank at most $r$ contains an $(m\cdot 2^{-O(\sqrt{r})})\times (n\cdot 2^{-O(\sqrt{r})})$ sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank $r$ is at most $O(\sqrt{r})$.
We consider the sparsification of sums $F : \mathbb{R}^n \to \mathbb{R}$ where $F(x) = f_1(\langle a_1,x\rangle) + \cdots + f_m(\langle a_m,x\rangle)$ for vectors $a_1,\ldots,a_m \in \mathbb{R}^n$ and functions $f_1,\ldots,f_m : \mathbb{R} \to \mathbb{R}_+$. We show that $(1+\varepsilon)$-approximate sparsifiers of $F$ with support size $\frac{n}{\varepsilon^2} (\log \frac{n}{\varepsilon})^{O(1)}$ exist whenever the functions $f_1,\ldots,f_m$ are symmetric, monotone, and satisfy natural growth bounds. Additionally, we give efficient algorithms to compute such a sparsifier assuming each $f_i$ can be evaluated efficiently. Our results generalize the classic case of $\ell_p$ sparsification, where $f_i(z) = |z|^p$, for $p \in (0, 2]$, and give the first near-linear size sparsifiers in the well-studied setting of the Huber loss function and its generalizations, e.g., $f_i(z) = \min\{|z|^p, |z|^2\}$ for $0 < p \leq 2$. Our sparsification algorithm can be applied to give near-optimal reductions for optimizing a variety of generalized linear models including $\ell_p$ regression for $p \in (1, 2]$ to high accuracy, via solving $(\log n)^{O(1)}$ sparse regression instances with $m \le n(\log n)^{O(1)}$, plus runtime proportional to the number of nonzero entries in the vectors $a_1, \dots, a_m$.
We study the statistical capacity of the classical binary perceptrons with general thresholds $\kappa$. After recognizing the connection between the capacity and the bilinearly indexed (bli) random processes, we utilize a recent progress in studying such processes to characterize the capacity. In particular, we rely on \emph{fully lifted} random duality theory (fl RDT) established in \cite{Stojnicflrdt23} to create a general framework for studying the perceptrons' capacities. Successful underlying numerical evaluations are required for the framework (and ultimately the entire fl RDT machinery) to become fully practically operational. We present results obtained in that directions and uncover that the capacity characterizations are achieved on the second (first non-trivial) level of \emph{stationarized} full lifting. The obtained results \emph{exactly} match the replica symmetry breaking predictions obtained through statistical physics replica methods in \cite{KraMez89}. Most notably, for the famous zero-threshold scenario, $\kappa=0$, we uncover the well known $\alpha\approx0.8330786$ scaled capacity.
A recent line of work has shown the unconditional advantage of constant-depth quantum computation, or $\mathsf{QNC^0}$, over $\mathsf{NC^0}$, $\mathsf{AC^0}$, and related models of classical computation. Problems exhibiting this advantage include search and sampling tasks related to the parity function, and it is natural to ask whether $\mathsf{QNC^0}$ can be used to help compute parity itself. We study $\mathsf{AC^0\circ QNC^0}$ -- a hybrid circuit model where $\mathsf{AC^0}$ operates on measurement outcomes of a $\mathsf{QNC^0}$ circuit, and conjecture $\mathsf{AC^0\circ QNC^0}$ cannot achieve $\Omega(1)$ correlation with parity. As evidence for this conjecture, we prove: $\bullet$ When the $\mathsf{QNC^0}$ circuit is ancilla-free, this model achieves only negligible correlation with parity. $\bullet$ For the general (non-ancilla-free) case, we show via a connection to nonlocal games that the conjecture holds for any class of postprocessing functions that has approximate degree $o(n)$ and is closed under restrictions, even when the $\mathsf{QNC^0}$ circuit is given arbitrary quantum advice. By known results this confirms the conjecture for linear-size $\mathsf{AC^0}$ circuits. $\bullet$ Towards a switching lemma for $\mathsf{AC^0\circ QNC^0}$, we study the effect of quantum preprocessing on the decision tree complexity of Boolean functions. We find that from this perspective, nonlocal channels are no better than randomness: a Boolean function $f$ precomposed with an $n$-party nonlocal channel is together equal to a randomized decision tree with worst-case depth at most $\mathrm{DT}_\mathrm{depth}[f]$. Our results suggest that while $\mathsf{QNC^0}$ is surprisingly powerful for search and sampling tasks, that power is "locked away" in the global correlations of its output, inaccessible to simple classical computation for solving decision problems.
Three variants of the statistical complexity function, which is used as a criterion in the problem of detection of a useful signal in the signal-noise mixture, are considered. The probability distributions maximizing the considered variants of statistical complexity are obtained analytically and conclusions about the efficiency of using one or another variant for detection problem are made. The comparison of considered information characteristics is shown and analytical results are illustrated on an example of synthesized signals. A method is proposed for selecting the threshold of the information criterion, which can be used in decision rule for useful signal detection in the signal-noise mixture. The choice of the threshold depends a priori on the analytically obtained maximum values. As a result, the complexity based on the total variation demonstrates the best ability of useful signal detection.
We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with energy density having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of 1-Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.
We study the optimal sample complexity of neighbourhood selection in linear structural equation models, and compare this to best subset selection (BSS) for linear models under general design. We show by example that -- even when the structure is \emph{unknown} -- the existence of underlying structure can reduce the sample complexity of neighbourhood selection. This result is complicated by the possibility of path cancellation, which we study in detail, and show that improvements are still possible in the presence of path cancellation. Finally, we support these theoretical observations with experiments. The proof introduces a modified BSS estimator, called klBSS, and compares its performance to BSS. The analysis of klBSS may also be of independent interest since it applies to arbitrary structured models, not necessarily those induced by a structural equation model. Our results have implications for structure learning in graphical models, which often relies on neighbourhood selection as a subroutine.
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