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In the standard formulation of the denoising problem, one is given a probabilistic model relating a latent variable $\Theta \in \Omega \subset \mathbb{R}^m \; (m\ge 1)$ and an observation $Z \in \mathbb{R}^d$ according to: $Z \mid \Theta \sim p(\cdot\mid \Theta)$ and $\Theta \sim G^*$, and the goal is to construct a map to recover the latent variable from the observation. The posterior mean, a natural candidate for estimating $\Theta$ from $Z$, attains the minimum Bayes risk (under the squared error loss) but at the expense of over-shrinking the $Z$, and in general may fail to capture the geometric features of the prior distribution $G^*$ (e.g., low dimensionality, discreteness, sparsity, etc.). To rectify these drawbacks, in this paper we take a new perspective on this denoising problem that is inspired by optimal transport (OT) theory and use it to propose a new OT-based denoiser at the population level setting. We rigorously prove that, under general assumptions on the model, our OT-based denoiser is well-defined and unique, and is closely connected to solutions to a Monge OT problem. We then prove that, under appropriate identifiability assumptions on the model, our OT-based denoiser can be recovered solely from information of the marginal distribution of $Z$ and the posterior mean of the model, after solving a linear relaxation problem over a suitable space of couplings that is reminiscent of a standard multimarginal OT (MOT) problem. In particular, thanks to Tweedie's formula, when the likelihood model $\{ p(\cdot \mid \theta) \}_{\theta \in \Omega}$ is an exponential family of distributions, the OT-based denoiser can be recovered solely from the marginal distribution of $Z$. In general, our family of OT-like relaxations is of interest in its own right and for the denoising problem suggests alternative numerical methods inspired by the rich literature on computational OT.

The two-parameter Mittag-Leffler function $E_{\alpha, \beta}$ is of fundamental importance in fractional calculus. It appears frequently in the solutions of fractional differential and integral equations. Nonetheless, this vital function is often expensive to compute. Several attempts have been made to construct cost-effective and accurate approximations. These attempts focus mainly on the completely monotone Mittag-Leffler functions. However, when $\alpha > 1$ the monotonicity property is largely lost and as such roots and oscillations are exhibited. Consequently, existing approximants constructed mainly for $\alpha \in (0,1)$ often fail to capture this oscillatory behavior. In this paper, we construct computationally efficient and accurate rational approximants for $E_{\alpha, \beta}(-t)$, $t \ge 0$, with $\alpha \in (1,2)$. This construction is fundamentally based on the decomposition of Mittag-Leffler function with real roots into one without and a polynomial. Following which new approximants are constructed by combining the global Pad\'e approximation with a polynomial of appropriate degree. The rational approximants are extended to approximation of matrix Mittag-Leffler and different approaches to achieve efficient implementation for matrix arguments are discussed. Numerical experiments are provided to illustrate the significant accuracy improvement achieved by the proposed approximants.

Let $X = \{X_{u}\}_{u \in U}$ be a real-valued Gaussian process indexed by a set $U$. It can be thought of as an undirected graphical model with every random variable $X_{u}$ serving as a vertex. We characterize this graph in terms of the covariance of $X$ through its reproducing kernel property. Unlike other characterizations in the literature, our characterization does not restrict the index set $U$ to be finite or countable, and hence can be used to model the intrinsic dependence structure of stochastic processes in continuous time/space. Consequently, this characterization is not in terms of the zero entries of an inverse covariance. This poses novel challenges for the problem of recovery of the dependence structure from a sample of independent realizations of $X$, also known as structure estimation. We propose a methodology that circumvents these issues, by targeting the recovery of the underlying graph up to a finite resolution, which can be arbitrarily fine and is limited only by the available sample size. The recovery is shown to be consistent so long as the graph is sufficiently regular in an appropriate sense. We derive corresponding convergence rates and finite sample guarantees. Our methodology is illustrated by means of a simulation study and two data analyses.

A cap set is a subset of $\mathbb{F}_3^n$ with no solutions to $x+y+z=0$ other than when $x=y=z$. In this paper, we provide a new lower bound on the size of a maximal cap set. Building on a construction of Edel, we use improved computational methods and new theoretical ideas to show that, for large enough $n$, there is always a cap set in $\mathbb{F}_3^n$ of size at least $2.218^n$.

We consider the problem of learning a graph modeling the statistical relations of the $d$ variables from a dataset with $n$ samples $X \in \mathbb{R}^{n \times d}$. Standard approaches amount to searching for a precision matrix $\Theta$ representative of a Gaussian graphical model that adequately explains the data. However, most maximum likelihood-based estimators usually require storing the $d^{2}$ values of the empirical covariance matrix, which can become prohibitive in a high-dimensional setting. In this work, we adopt a compressive viewpoint and aim to estimate a sparse $\Theta$ from a \emph{sketch} of the data, i.e. a low-dimensional vector of size $m \ll d^{2}$ carefully designed from $X$ using non-linear random features. Under certain assumptions on the spectrum of $\Theta$ (or its condition number), we show that it is possible to estimate it from a sketch of size $m=\Omega\left((d+2k)\log(d)\right)$ where $k$ is the maximal number of edges of the underlying graph. These information-theoretic guarantees are inspired by compressed sensing theory and involve restricted isometry properties and instance optimal decoders. We investigate the possibility of achieving practical recovery with an iterative algorithm based on the graphical lasso, viewed as a specific denoiser. We compare our approach and graphical lasso on synthetic datasets, demonstrating its favorable performance even when the dataset is compressed.

Given a collection of $m$ sets from a universe $\mathcal{U}$, the Maximum Set Coverage problem consists of finding $k$ sets whose union has largest cardinality. This problem is NP-Hard, but the solution can be approximated by a polynomial time algorithm up to a factor $1-1/e$. However, this algorithm does not scale well with the input size. In a streaming context, practical high-quality solutions are found, but with space complexity that scales linearly with respect to the size of the universe $|\mathcal{U}|$. However, one randomized streaming algorithm has been shown to produce a $1-1/e-\varepsilon$ approximation of the optimal solution with a space complexity that scales only poly-logarithmically with respect to $m$ and $|\mathcal{U}|$. In order to achieve such a low space complexity, the authors used a technique called subsampling, based on independent-wise hash functions, and $F_0$-sketching. This article focuses on this sublinear-space algorithm and introduces methods to reduce the time cost of subsampling. Firstly, we give some optimizations that do not alter the space complexity, number of passes and approximation quality of the original algorithm. In particular, we reanalyze the error bounds to show that the original independence factor of $\Omega(\varepsilon^{-2} k \log m)$ can be fine-tuned to $\Omega(k \log m)$. Secondly we show that $F_0$-sketching can be replaced by a much more simple mechanism. Finally, our experimental results show that even a pairwise-independent hash-function sampler does not produce worse solution than the original algorithm, while running significantly faster by several orders of magnitude.

We propose a test of many zero parameter restrictions in a high dimensional linear iid regression model with $k$ $>>$ $n$ regressors. The test statistic is formed by estimating key parameters one at a time based on many low dimension regression models with nuisance terms. The parsimoniously parametrized models identify whether the original parameter of interest is or is not zero. Estimating fixed low dimension sub-parameters ensures greater estimator accuracy, it does not require a sparsity assumption nor therefore a regularized estimator, it is computationally fast compared to, e.g., de-biased Lasso, and using only the largest in a sequence of weighted estimators reduces test statistic complexity and therefore estimation error. We provide a parametric wild bootstrap for p-value computation, and prove the test is consistent and has non-trivial $\sqrt{n/\{\ln (n)\mathcal{M}% _{n}\}}$-local-to-null power where $\mathcal{M}_{n}$ is the $l_{\infty }$ covariate fourth moment.

This paper studies robust nonparametric regression, in which an adversarial attacker can modify the values of up to $q$ samples from a training dataset of size $N$. Our initial solution is an M-estimator based on Huber loss minimization. Compared with simple kernel regression, i.e. the Nadaraya-Watson estimator, this method can significantly weaken the impact of malicious samples on the regression performance. We provide the convergence rate as well as the corresponding minimax lower bound. The result shows that, with proper bandwidth selection, $\ell_\infty$ error is minimax optimal. The $\ell_2$ error is optimal with relatively small $q$, but is suboptimal with larger $q$. The reason is that this estimator is vulnerable if there are many attacked samples concentrating in a small region. To address this issue, we propose a correction method by projecting the initial estimate to the space of Lipschitz functions. The final estimate is nearly minimax optimal for arbitrary $q$, up to a $\ln N$ factor.

We consider the Distinct Shortest Walks problem. Given two vertices $s$ and $t$ of a graph database $\mathcal{D}$ and a regular path query, enumerate all walks of minimal length from $s$ to $t$ that carry a label that conforms to the query. Usual theoretical solutions turn out to be inefficient when applied to graph models that are closer to real-life systems, in particular because edges may carry multiple labels. Indeed, known algorithms may repeat the same answer exponentially many times. We propose an efficient algorithm for multi-labelled graph databases. The preprocessing runs in $O{|\mathcal{D}|\times|\mathcal{A}|}$ and the delay between two consecutive outputs is in $O(\lambda\times|\mathcal{A}|)$, where $\mathcal{A}$ is a nondeterministic automaton representing the query and $\lambda$ is the minimal length. The algorithm can handle $\varepsilon$-transitions in $\mathcal{A}$ or queries given as regular expressions at no additional cost.

Quantum copy protection, introduced by Aaronson, enables giving out a quantum program-description that cannot be meaningfully duplicated. Despite over a decade of study, copy protection is only known to be possible for a very limited class of programs. As our first contribution, we show how to achieve "best-possible" copy protection for all programs. We do this by introducing quantum state indistinguishability obfuscation (qsiO), a notion of obfuscation for quantum descriptions of classical programs. We show that applying qsiO to a program immediately achieves best-possible copy protection. Our second contribution is to show that, assuming injective one-way functions exist, qsiO is concrete copy protection for a large family of puncturable programs -- significantly expanding the class of copy-protectable programs. A key tool in our proof is a new variant of unclonable encryption (UE) that we call coupled unclonable encryption (cUE). While constructing UE in the standard model remains an important open problem, we are able to build cUE from one-way functions. If we additionally assume the existence of UE, then we can further expand the class of puncturable programs for which qsiO is copy protection. Finally, we construct qsiO relative to an efficient quantum oracle.