Penalized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Often, the penalties are geometrically decomposable, i.e. can be expressed as a sum of support functions over convex sets. We generalize the notion of irrepresentable to geometrically decomposable penalties and develop a general framework for establishing consistency and model selection consistency of M-estimators with such penalties. We then use this framework to derive results for some special cases of interest in bioinformatics and statistical learning.

This work studies the expressivity of ReLU neural networks with a focus on their depth. A sequence of previous works showed that $\lceil \log_2(n+1) \rceil$ hidden layers are sufficient to compute all continuous piecewise linear (CPWL) functions on $\mathbb{R}^n$. Hertrich, Basu, Di Summa, and Skutella (NeurIPS'21 / SIDMA'23) conjectured that this result is optimal in the sense that there are CPWL functions on $\mathbb{R}^n$, like the maximum function, that require this depth. We disprove the conjecture and show that $\lceil\log_3(n-1)\rceil+1$ hidden layers are sufficient to compute all CPWL functions on $\mathbb{R}^n$. A key step in the proof is that ReLU neural networks with two hidden layers can exactly represent the maximum function of five inputs. More generally, we show that $\lceil\log_3(n-2)\rceil+1$ hidden layers are sufficient to compute the maximum of $n\geq 4$ numbers. Our constructions almost match the $\lceil\log_3(n)\rceil$ lower bound of Averkov, Hojny, and Merkert (ICLR'25) in the special case of ReLU networks with weights that are decimal fractions. The constructions have a geometric interpretation via polyhedral subdivisions of the simplex into ``easier'' polytopes.

The well-known Condorcet Jury Theorem states that, under majority rule, the better of two alternatives is chosen with probability approaching one as the population grows. We study an asymmetric setting where voters face varying participation costs and share a possibly heuristic belief about their pivotality (ability to influence the outcome). In a costly voting setup where voters abstain if their participation cost is greater than their pivotality estimate, we identify a single property of the heuristic belief -- weakly vanishing pivotality -- that gives rise to multiple stable equilibria in which elections are nearly tied. In contrast, strongly vanishing pivotality (as in the standard Calculus of Voting model) yields a unique, trivial equilibrium where only zero-cost voters participate as the population grows. We then characterize when nontrivial equilibria satisfy a version of the Jury Theorem: below a sharp threshold, the majority-preferred candidate wins with probability approaching one; above it, both candidates either win with equal probability.

Neural Information Processing Systems http://nips.cc/

Social media echo chambers play a central role in the spread of misinformation, yet existing models often overlook the influence of individual confirmation bias. An existing model of echo chambers is the "gravity well" model, which creates an analog between echo chambers and spatial gravity wells. We extend this established model by introducing a dynamic confirmation bias variable that adjusts the strength of pull based on a user's susceptibility to belief-reinforcing content. This variable is calculated for each user through comparisons between their posting history and their responses to posts of a wide range of viewpoints. Incorporating this factor produces a confirmation-bias-integrated gravity well model that more accurately identifies echo chambers and reveals community-level markers of information health. We validated the approach on nineteen Reddit communities, demonstrating improved detection of echo chambers. Our contribution is a framework for systematically capturing the role of confirmation bias in online group dynamics, enabling more effective identification of echo chambers. By flagging these high-risk environments, the model supports efforts to curb the spread of misinformation at its most common points of amplification.

The main object of interest in the present work is the convex floating body . Let us first recall the definition of the convex floating body. The floating body has the following desirable properties. The floating body is a natural high dimensional statistical construction. Approximate Differential Privacy Throughout the paper we referred to a similar notion to "pure" In this Section we establish our main meta-theorem, Theorem 10.

Electrical impedance tomography (EIT) is a non-invasive imaging method with diverse applications, including medical imaging and non-destructive testing. The inverse problem of reconstructing internal electrical conductivity from boundary measurements is nonlinear and highly ill-posed, making it difficult to solve accurately. In recent years, there has been growing interest in combining analytical methods with machine learning to solve inverse problems. In this paper, we propose a method for estimating the convex hull of inclusions from boundary measurements by combining the enclosure method proposed by Ikehata with neural networks. We demonstrate its performance using experimental data. Compared to the classical enclosure method with least squares fitting, the learned convex hull achieves superior performance on both simulated and experimental data.

Forp = 3, this covers the cases of binary fractions as well as decimal fractions, two of the most common practical settings. Moreover, it shows that the expressive power of ReLU networks grows for every N up to O(logn). In the case of rational weights that are N-ary fractions for any fixed N, even allowing arbitrarily large denominators and arbitrary width does not facilitate exact representations of low constant depth. Theorem 4 can be viewed as a partial confirmation of Conjecture 1 for rational weights, as the term lnlnN is growing extremely slowly in N. If one could replace lnlnN by a constant, the conjecture would be confirmed for rational weights, up to a constant multiple. As already highlighted in Haase et al. (2023), confirmation of the conjecture would theoretically explain the significance of max-pooling in the context of ReLU networks: It seems that the expressive power of ReLU is not enough to model the maximum of a large number of input variables unless network architectures of high-enough depth are used. So, enhancing ReLU networks with max-pooling layers could be a way to reach higher expressive power with shallow networks.

The well-known Condorcet's Jury theorem posits that the majority rule selects the best alternative among two available options with probability one, as the population size increases to infinity. We study this result under an asymmetric two-candidate setup, where supporters of both candidates may have different participation costs. When the decision to abstain is fully rational i.e., when the vote pivotality is the probability of a tie, the only equilibrium outcome is a trivial equilibrium where all voters except those with zero voting cost, abstain. We propose and analyze a more practical, boundedly rational model where voters overestimate their pivotality, and show that under this model, non-trivial equilibria emerge where the winning probability of both candidates is bounded away from one. We show that when the pivotality estimate strongly depends on the margin of victory, victory is not assured to any candidate in any non-trivial equilibrium, regardless of population size and in contrast to Condorcet's assertion. Whereas, under a weak dependence on margin, Condorcet's Jury theorem is restored.