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The dynamic ranking, due to its increasing importance in many applications, is becoming crucial, especially with the collection of voluminous time-dependent data. One such application is sports statistics, where dynamic ranking aids in forecasting the performance of competitive teams, drawing on historical and current data. Despite its usefulness, predicting and inferring rankings pose challenges in environments necessitating time-dependent modeling. This paper introduces a spectral ranker called Kernel Rank Centrality, designed to rank items based on pairwise comparisons over time. The ranker operates via kernel smoothing in the Bradley-Terry model, utilizing a Markov chain model. Unlike the maximum likelihood approach, the spectral ranker is nonparametric, demands fewer model assumptions and computations, and allows for real-time ranking. We establish the asymptotic distribution of the ranker by applying an innovative group inverse technique, resulting in a uniform and precise entrywise expansion. This result allows us to devise a new inferential method for predictive inference, previously unavailable in existing approaches. Our numerical examples showcase the ranker's utility in predictive accuracy and constructing an uncertainty measure for prediction, leveraging data from the National Basketball Association (NBA). The results underscore our method's potential compared to the gold standard in sports, the Arpad Elo rating system.

In this paper, we propose a multidimensional statistical model of intraday electricity prices at the scale of the trading session, which allows all products to be simulated simultaneously. This model, based on Poisson measures and inspired by the Common Shock Poisson Model, reproduces the Samuelson effect (intensity and volatility increases as time to maturity decreases). It also reproduces the price correlation structure, highlighted here in the data, which decreases as two maturities move apart. This model has only three parameters that can be estimated using a moment method that we propose here. We demonstrate the usefulness of the model on a case of storage valuation by dynamic programming over a trading session.

This note addresses the question of optimally estimating a linear functional of an object acquired through linear observations corrupted by random noise, where optimality pertains to a worst-case setting tied to a symmetric, convex, and closed model set containing the object. It complements the article "Statistical Estimation and Optimal Recovery" published in the Annals of Statistics in 1994. There, Donoho showed (among other things) that, for Gaussian noise, linear maps provide near-optimal estimation schemes relatively to a performance measure relevant in Statistical Estimation. Here, we advocate for a different performance measure arguably more relevant in Optimal Recovery. We show that, relatively to this new measure, linear maps still provide near-optimal estimation schemes even if the noise is merely log-concave. Our arguments, which make a connection to the deterministic noise situation and bypass properties specific to the Gaussian case, offer an alternative to parts of Donoho's proof.

In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to the closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on metrics. In this paper, we give a deterministic decremental $(2+\epsilon)$-approximation algorithm and a randomized incremental $(4+\epsilon)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+\epsilon)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art upper bound for maintaining $(1+\epsilon)$-approximate single-source distances in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+\epsilon)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances.

A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based representations, the misfit is instead minimized in a small subspace spanned by the first few eigenfunctions of a judicious elliptic operator, which itself depends on the previous iteration. By repeatedly adapting both the dimension and the basis of the search space, regularization is inherently incorporated at each iteration without the need for extra Tikhonov penalization. Convergence is proved under an angle condition, which is included into the resulting \emph{Adaptive Spectral Inversion} (ASI) algorithm. The ASI approach compares favorably to standard grid-based inversion using $L^2$-Tikhonov regularization when applied to an elliptic inverse problem. The improved accuracy resulting from the newly included angle condition is further demonstrated via numerical experiments from time-dependent inverse scattering problems.

State-of-the-art neural networks require extreme computational power to train. It is therefore natural to wonder whether they are optimally trained. Here we apply a recent advancement in stochastic thermodynamics which allows bounding the speed at which one can go from the initial weight distribution to the final distribution of the fully trained network, based on the ratio of their Wasserstein-2 distance and the entropy production rate of the dynamical process connecting them. Considering both gradient-flow and Langevin training dynamics, we provide analytical expressions for these speed limits for linear and linearizable neural networks e.g. Neural Tangent Kernel (NTK). Remarkably, given some plausible scaling assumptions on the NTK spectra and spectral decomposition of the labels -- learning is optimal in a scaling sense. Our results are consistent with small-scale experiments with Convolutional Neural Networks (CNNs) and Fully Connected Neural networks (FCNs) on CIFAR-10, showing a short highly non-optimal regime followed by a longer optimal regime.

Latent linear dynamical systems with Bernoulli observations provide a powerful modeling framework for identifying the temporal dynamics underlying binary time series data, which arise in a variety of contexts such as binary decision-making and discrete stochastic processes (e.g., binned neural spike trains). Here we develop a spectral learning method for fast, efficient fitting of probit-Bernoulli latent linear dynamical system (LDS) models. Our approach extends traditional subspace identification methods to the Bernoulli setting via a transformation of the first and second sample moments. This results in a robust, fixed-cost estimator that avoids the hazards of local optima and the long computation time of iterative fitting procedures like the expectation-maximization (EM) algorithm. In regimes where data is limited or assumptions about the statistical structure of the data are not met, we demonstrate that the spectral estimate provides a good initialization for Laplace-EM fitting. Finally, we show that the estimator provides substantial benefits to real world settings by analyzing data from mice performing a sensory decision-making task.

While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the LeadingOnes and Jump benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation $\sigma$ into another one $\tau$, but also the precise cycle structure of $\sigma \tau^{-1}$. For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of $\Theta(n)$. Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $m^{\Theta(m)}$ on jump functions with jump size $m$. A short empirical analysis confirms these findings, but also reveals that small implementation details like the rate of void mutations can make an important difference.

Supervised classification algorithms are used to solve a growing number of real-life problems around the globe. Their performance is strictly connected with the quality of labels used in training. Unfortunately, acquiring good-quality annotations for many tasks is infeasible or too expensive to be done in practice. To tackle this challenge, active learning algorithms are commonly employed to select only the most relevant data for labeling. However, this is possible only when the quality and quantity of labels acquired from experts are sufficient. Unfortunately, in many applications, a trade-off between annotating individual samples by multiple annotators to increase label quality vs. annotating new samples to increase the total number of labeled instances is necessary. In this paper, we address the issue of faulty data annotations in the context of active learning. In particular, we propose two novel annotation unification algorithms that utilize unlabeled parts of the sample space. The proposed methods require little to no intersection between samples annotated by different experts. Our experiments on four public datasets indicate the robustness and superiority of the proposed methods in both, the estimation of the annotator's reliability, and the assignment of actual labels, against the state-of-the-art algorithms and the simple majority voting.

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.