Let $X$ be a $p$-variate random vector and $\widetilde{X}$ a knockoff copy of $X$ (in the sense of \cite{CFJL18}). A new approach for constructing $\widetilde{X}$ (henceforth, NA) has been introduced in \cite{JSPI}. NA has essentially three advantages: (i) To build $\widetilde{X}$ is straightforward; (ii) The joint distribution of $(X,\widetilde{X})$ can be written in closed form; (iii) $\widetilde{X}$ is often optimal under various criteria. However, for NA to apply, $X_1,\ldots, X_p$ should be conditionally independent given some random element $Z$. Our first result is that any probability measure $\mu$ on $\mathbb{R}^p$ can be approximated by a probability measure $\mu_0$ of the form $$\mu_0\bigl(A_1\times\ldots\times A_p\bigr)=E\Bigl\{\prod_{i=1}^p P(X_i\in A_i\mid Z)\Bigr\}.$$ The approximation is in total variation distance when $\mu$ is absolutely continuous, and an explicit formula for $\mu_0$ is provided. If $X\sim\mu_0$, then $X_1,\ldots,X_p$ are conditionally independent. Hence, with a negligible error, one can assume $X\sim\mu_0$ and build $\widetilde{X}$ through NA. Our second result is a characterization of the knockoffs $\widetilde{X}$ obtained via NA. It is shown that $\widetilde{X}$ is of this type if and only if the pair $(X,\widetilde{X})$ can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti's theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of $\widetilde{X}$ given $X$ is obtained in a few cases. In one of such cases, it is assumed $X_i\in\{0,1\}$ for all $i$.
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This work concerns elementwise-transformations of spiked matrices: $Y_n = n^{-1/2} f(n^{1-1/(2\ell_*)} X_n + Z_n)$. Here, $f$ is a function applied elementwise, $X_n$ is a low-rank signal matrix, $Z_n$ is white noise, and $\ell_* \geq 1$ is an integer. We find that principal component analysis is powerful for recovering low-rank signal under highly non-linear and discontinuous transformations. Specifically, in the high-dimensional setting where $Y_n$ is of size $n \times p$ with $n,p \rightarrow \infty$ and $p/n \rightarrow \gamma \in (0, \infty)$, we uncover a phase transition: for signal-to-noise ratios above a sharp threshold -- depending on $f$, the distribution of elements of $Z_n$, and the limiting aspect ratio $\gamma$ -- the principal components of $Y_n$ (partially) recover those of $X_n$. Below this threshold, the principal components are asymptotically orthogonal to the signal. In contrast, in the standard setting where $X_n + n^{-1/2}Z_n$ is observed directly, the analogous phase transition depends only on $\gamma$. Analogous phenomena occur with $X_n$ square and symmetric and $Z_n$ a generalized Wigner matrix.
A linear code $C$ over $\mathbb{F}_q$ is called $\Delta$-divisible if the Hamming weights $\operatorname{wt}(c)$ of all codewords $c \in C$ are divisible by $\Delta$. The possible effective lengths of $q^r$-divisible codes have been completely characterized for each prime power $q$ and each non-negative integer $r$. The study of $\Delta$ divisible codes was initiated by Harold Ward. If $c$ divides $\Delta$ but is coprime to $q$, then each $\Delta$-divisible code $C$ over $\F_q$ is the $c$-fold repetition of a $\Delta/c$-divisible code. Here we determine the possible effective lengths of $p^r$-divisible codes over finite fields of characteristic $p$, where $p\in\mathbb{N}$ but $p^r$ is not a power of the field size, i.e., the missing cases.
The contraction$^*$-depth is the matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph $G$ is the minimum height of a rooted forest whose closure contains $G$ by proving the following for every matroid $M$ (except the trivial case when $M$ consists of loops and bridges only): the contraction$^*$-depth of $M$ plus one is equal to the minimum contraction-depth of a matroid containing $M$ as a restriction.
A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for each $i$. An independent set $I$ in $H$ is balanced if $|I\cap V_i| = |I|/k$ for each $i$, and a coloring is balanced if each color class induces a balanced independent set in $H$. In this paper, we provide a lower bound on the balanced independence number $\alpha_b(H)$ in terms of the average degree $D = |E|/n$, and an upper bound on the balanced chromatic number $\chi_b(H)$ in terms of the maximum degree $\Delta$. Our results match those of recent work of Chakraborti for $k = 2$.
Quantifying the difference between two probability density functions, $p$ and $q$, using available data, is a fundamental problem in Statistics and Machine Learning. A usual approach for addressing this problem is the likelihood-ratio estimation (LRE) between $p$ and $q$, which -- to our best knowledge -- has been investigated mainly for the offline case. This paper contributes by introducing a new framework for online non-parametric LRE (OLRE) for the setting where pairs of iid observations $(x_t \sim p, x'_t \sim q)$ are observed over time. The non-parametric nature of our approach has the advantage of being agnostic to the forms of $p$ and $q$. Moreover, we capitalize on the recent advances in Kernel Methods and functional minimization to develop an estimator that can be efficiently updated online. We provide theoretical guarantees for the performance of the OLRE method along with empirical validation in synthetic experiments.
In this paper we propose a definition of the distributional Riemann curvature tensor in dimension $N\geq 2$ if the underlying metric tensor $g$ defined on a triangulation $\mathcal{T}$ possesses only single-valued tangential-tangential components on codimension 1 simplices. We analyze the convergence of the curvature approximation in the $H^{-2}$-norm if a sequence of interpolants $g_h$ of polynomial order $k\geq 0$ of a smooth metric $g$ is given. We show that for dimension $N=2$ convergence rates of order $\mathcal{O}(h^{k+1})$ are obtained. For $N\geq 3$ convergence holds only in the case $k\geq 1$. Numerical examples demonstrate that our theoretical results are sharp. By choosing appropriate test functions we show that the distributional Gauss and scalar curvature in 2D respectively any dimension are obtained. Further, a first definition of the distributional Ricci curvature tensor in arbitrary dimension is derived, for which our analysis is applicable.
When the unknown regression function of a single variable is known to have derivatives up to the $(\gamma+1)$th order bounded in absolute values by a common constant everywhere or a.e. (i.e., $(\gamma+1)$th degree of smoothness), the minimax optimal rate of the mean integrated squared error (MISE) is stated as $\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ in the literature. This paper shows that: (i) if $n\leq\left(\gamma+1\right)^{2\gamma+3}$, the minimax optimal MISE rate is $\frac{\log n}{n\log(\log n)}$ and the optimal degree of smoothness to exploit is roughly $\max\left\{ \left\lfloor \frac{\log n}{2\log\left(\log n\right)}\right\rfloor ,\,1\right\} $; (ii) if $n>\left(\gamma+1\right)^{2\gamma+3}$, the minimax optimal MISE rate is $\left(\frac{1}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ and the optimal degree of smoothness to exploit is $\gamma+1$. The fundamental contribution of this paper is a set of metric entropy bounds we develop for smooth function classes. Some of our bounds are original, and some of them improve and/or generalize the ones in the literature (e.g., Kolmogorov and Tikhomirov, 1959). Our metric entropy bounds allow us to show phase transitions in the minimax optimal MISE rates associated with some commonly seen smoothness classes as well as non-standard smoothness classes, and can also be of independent interest outside the nonparametric regression problems.
Due to the lack of a canonical ordering in ${\mathbb R}^d$ for $d>1$, defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been shown to offer a solution in which multivariate ranks are obtained by transporting data points to a grid that approximates a uniform reference measure (Chernozhukov et al., 2017; Hallin, 2017; Hallin et al., 2021), thereby inducing ranks, signs, and a data-driven ordering of ${\mathbb R}^d$. We take up this new perspective to define and study multivariate analogues of the sign covariance/quadrant statistic, Spearman's rho, Kendall's tau, and van der Waerden covariances. The resulting tests of multivariate independence are fully distribution-free, hence uniformly valid irrespective of the actual (absolutely continuous) distribution of the observations. Our results provide the asymptotic distribution theory for these new test statistics, with asymptotic approximations to critical values to be used for testing independence between random vectors, as well as a power analysis of the resulting tests in an extension of the so-called Konijn model. For the van der Waerden tests, this power analysis includes a multivariate Chernoff--Savage property guaranteeing that, under elliptical generalized Konijn models, the asymptotic relative efficiency with respect to Wilks' classical (pseudo-)Gaussian procedure of our van der Waerden tests is strictly larger than or equal to one, where equality is achieved under Gaussian distributions only. We similarly provide a lower bound for the asymptotic relative efficiency of our Spearman procedure with respect to Wilks' test, thus extending the classical result by Hodges and Lehmann on the asymptotic relative efficiency, in univariate location models, of Wilcoxon tests with respect to the Student ones.
We present a novel stochastic variational Gaussian process ($\mathcal{GP}$) inference method, based on a posterior over a learnable set of weighted pseudo input-output points (coresets). Instead of a free-form variational family, the proposed coreset-based, variational tempered family for $\mathcal{GP}$s (CVTGP) is defined in terms of the $\mathcal{GP}$ prior and the data-likelihood; hence, accommodating the modeling inductive biases. We derive CVTGP's lower bound for the log-marginal likelihood via marginalization of the proposed posterior over latent $\mathcal{GP}$ coreset variables, and show it is amenable to stochastic optimization. CVTGP reduces the learnable parameter size to $\mathcal{O}(M)$, enjoys numerical stability, and maintains $\mathcal{O}(M^3)$ time- and $\mathcal{O}(M^2)$ space-complexity, by leveraging a coreset-based tempered posterior that, in turn, provides sparse and explainable representations of the data. Results on simulated and real-world regression problems with Gaussian observation noise validate that CVTGP provides better evidence lower-bound estimates and predictive root mean squared error than alternative stochastic $\mathcal{GP}$ inference methods.
We investigate the relation between $\delta$ and $\epsilon$ required for obtaining a $(1+\delta)$-approximation in time $N^{2-\epsilon}$ for closest pair problems under various distance metrics, and for other related problems in fine-grained complexity. Specifically, our main result shows that if it is impossible to (exactly) solve the (bichromatic) inner product (IP) problem for vectors of dimension $c \log N$ in time $N^{2-\epsilon}$, then there is no $(1+\delta)$-approximation algorithm for (bichromatic) Euclidean Closest Pair running in time $N^{2-2\epsilon}$, where $\delta \approx (\epsilon/c)^2$ (where $\approx$ hides $\polylog$ factors). This improves on the prior result due to Chen and Williams (SODA 2019) which gave a smaller polynomial dependence of $\delta$ on $\epsilon$, on the order of $\delta \approx (\epsilon/c)^6$. Our result implies in turn that no $(1+\delta)$-approximation algorithm exists for Euclidean closest pair for $\delta \approx \epsilon^4$, unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of $\delta \approx \epsilon^3$ for Euclidean closest pair, given by the best known algorithm of Almam, Chan, and Williams (FOCS 2016). By known reductions, a similar result follows for a host of other related problems in fine-grained hardness of approximation. Our reduction combines the hardness of approximation framework of Chen and Williams, together with an MA communication protocol for IP over a small alphabet, that is inspired by the MA protocol of Chen (Theory of Computing, 2020).
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