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In many recommender systems and search problems, presenting a well balanced set of results can be an important goal in addition to serving highly relevant content. For example, in a movie recommendation system, it may be helpful to achieve a certain balance of different genres, likewise, it may be important to balance between highly popular versus highly personalized shows. Such balances could be thought across many categories and may be required for enhanced user experience, business considerations, fairness objectives etc. In this paper, we consider the problem of calibrating with respect to any given categories over items. We propose a way to balance a trade-off between relevance and calibration via a Linear Programming optimization problem where we learn a doubly stochastic matrix to achieve optimal balance in expectation. We then realize the learned policy using the Birkhoff-von Neumann decomposition of a doubly stochastic matrix. Several optimizations are considered over the proposed basic approach to make it fast. The experiments show that the proposed formulation can achieve a much better trade-off compared to many other baselines. This paper does not prescribe the exact categories to calibrate over (such as genres) universally for applications. This is likely dependent on the particular task or business objective. The main contribution of the paper is that it proposes a framework that can be applied to a variety of problems and demonstrates the efficacy of the proposed method using a few use-cases.

Noiseless compressive sensing is a protocol that enables undersampling and later recovery of a signal without loss of information. This compression is possible because the signal is usually sufficiently sparse in a given basis. Currently, the algorithm offering the best tradeoff between compression rate, robustness, and speed for compressive sensing is the LASSO (l1-norm bias) algorithm. However, many studies have pointed out the possibility that the implementation of lp-norms biases, with p smaller than one, could give better performance while sacrificing convexity. In this work, we focus specifically on the extreme case of the l0-based reconstruction, a task that is complicated by the discontinuity of the loss. In the first part of the paper, we describe via statistical physics methods, and in particular the replica method, how the solutions to this optimization problem are arranged in a clustered structure. We observe two distinct regimes: one at low compression rate where the signal can be recovered exactly, and one at high compression rate where the signal cannot be recovered accurately. In the second part, we present two message-passing algorithms based on our first results for the l0-norm optimization problem. The proposed algorithms are able to recover the signal at compression rates higher than the ones achieved by LASSO while being computationally efficient.

Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning. Instead of deriving guarantees on the usual estimation error, the goal of this paper is to provide concentration inequalities on the distance between the sets of minimizers of the risks for a broad spectrum of estimation problems. In particular, the risks are defined on metric spaces through probability measures that are also supported on metric spaces. A particular attention will therefore be given to include unbounded spaces and non-convex cost functions that might also be unbounded. This work identifies a set of assumptions allowing to describe a regime that seem to govern the concentration in many estimation problems, where the empirical minimizers are stable. This stability can then be leveraged to prove parametric concentration rates in probability and in expectation. The assumptions are verified, and the bounds showcased, on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters.

In this paper we obtain quantitative {\it Bernstein-von Mises type} bounds on the normal approximation of the posterior distribution in exponential family models when centering either around the posterior mode or around the maximum likelihood estimator. Our bounds, obtained through a version of Stein's method, are non-asymptotic, and data dependent; they are of the correct order both in the total variation and Wasserstein distances, as well as for approximations for expectations of smooth functions of the posterior. All our results are valid for univariate and multivariate posteriors alike, and do not require a conjugate prior setting. We illustrate our findings on a variety of exponential family distributions, including Poisson, multinomial and normal distribution with unknown mean and variance. The resulting bounds have an explicit dependence on the prior distribution and on sufficient statistics of the data from the sample, and thus provide insight into how these factors may affect the quality of the normal approximation.

Polarization-adjusted convolutional (PAC) codes can approach the theoretical bound for block error rate (BLER) performance at short-to-medium codeword length. PAC codes have excellent BLER performance using Monte Carlo (MC) rate-profiles and Weighted Sum (WS) rate-profiles, but the BLER performances of the constructed codes still fall away from the dispersion bound at high signal-to-noise ratios (SNR). This paper proposes a List-Search (LS) construction method for PAC codes, which considers the influence of weight spectrum on BLER performance and the condition that sequence decoding for PAC codes having a finite mean computational complexity. The proposed construction method using LS can reduce the number of minimum weight codewords of PAC codes. The BLER performance of the constructed codes is better than that of the constructed codes using MC rate-profiles or WS rate-profiles, and can approach the dispersion bound at high SNR. Moreover, the BLER performance of successive cancellation list (SCL) decoding PAC codes using LS rate-profiles can approach the theoretical bound, but SCL decoding requires a large number of sorting operations. To reduce the number of sorting operations, a path-splitting critical sets (PSCS) construction method is proposed. The PSCS obtained by this method are the information bits subset that have the greatest influence on the number of minimum weight codewords. The simulation results show that this method can significantly reduce the number of sorting operations during SCL-type decoding.

Moves in chess games are usually analyzed on a case-by-case basis by professional players, but thanks to the availability of large game databases, we can envision another approach of the game. Here, we indeed adopt a very different point of view, and analyze moves in chess games from a statistical point of view. We first focus on spatial properties and the location of pieces and show that the number of possible moves during a game is positively correlated with its outcome. We then study heatmaps of pieces and show that the spatial distribution of pieces varies less between human players than with engines (such as Stockfish): engines seem to use pieces in a very different way as human did for centuries. These heatmaps also allow us to construct a distance between players that characterizes how they use their pieces. In a second part, we focus on the best move and the second best move found by Stockfish and study the difference $\Delta$ of their evaluation. We found different regimes during a chess game. In a `quiet' regime, $\Delta$ is small, indicating that many paths are possible for both players. In contrast, there are also `volatile' regimes characterized by a `tipping point', for which $\Delta$ becomes large. At these tipping points, the outcome could then switch completely depending on the move chosen. We also found that for a large number of games, the distribution of $\Delta$ can be fitted by a power law $P(\Delta)\sim \Delta^{-\beta}$ with an exponent that seems to be universal (for human players and engines) and around $\beta\approx 1.8$. The probability to encounter a tipping point in a game is therefore far from being negligible. Finally, we conclude by mentioning possible directions of research for a quantitative understanding of chess games such as the structure of the pawn chain, the interaction graph between pieces, or a quantitative definition of critical points.

The paper analyses properties of a large class of "path-based" Data Envelopment Analysis models through a unifying general scheme. The scheme includes the well-known oriented radial models, the hyperbolic distance function model, the directional distance function models, and even permits their generalisations. The modelling is not constrained to non-negative data and is flexible enough to accommodate variants of standard models over arbitrary data. Mathematical tools developed in the paper allow systematic analysis of the models from the point of view of ten desirable properties. It is shown that some of the properties are satisfied (resp., fail) for all models in the general scheme, while others have a more nuanced behaviour and must be assessed individually in each model. Our results can help researchers and practitioners navigate among the different models and apply the models to mixed data.

Evaluating the quality of learned representations without relying on a downstream task remains one of the challenges in representation learning. In this work, we present Geometric Component Analysis (GeomCA) algorithm that evaluates representation spaces based on their geometric and topological properties. GeomCA can be applied to representations of any dimension, independently of the model that generated them. We demonstrate its applicability by analyzing representations obtained from a variety of scenarios, such as contrastive learning models, generative models and supervised learning models.

We present self-supervised geometric perception (SGP), the first general framework to learn a feature descriptor for correspondence matching without any ground-truth geometric model labels (e.g., camera poses, rigid transformations). Our first contribution is to formulate geometric perception as an optimization problem that jointly optimizes the feature descriptor and the geometric models given a large corpus of visual measurements (e.g., images, point clouds). Under this optimization formulation, we show that two important streams of research in vision, namely robust model fitting and deep feature learning, correspond to optimizing one block of the unknown variables while fixing the other block. This analysis naturally leads to our second contribution -- the SGP algorithm that performs alternating minimization to solve the joint optimization. SGP iteratively executes two meta-algorithms: a teacher that performs robust model fitting given learned features to generate geometric pseudo-labels, and a student that performs deep feature learning under noisy supervision of the pseudo-labels. As a third contribution, we apply SGP to two perception problems on large-scale real datasets, namely relative camera pose estimation on MegaDepth and point cloud registration on 3DMatch. We demonstrate that SGP achieves state-of-the-art performance that is on-par or superior to the supervised oracles trained using ground-truth labels.

Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.