Related papers: What's Up with Downward Collapse: Using the Easy-H…

This work reports deep-learning-unique first-order and second-order phase transitions, whose phenomenology closely follows that in statistical physics. In particular, we prove that the competition between prediction error and model…

This chapter provides a hands-on tutorial on the important technique known as self-reducibility. Through a series of "Challenge Problems" that are theorems that the reader will---after being given definitions and tools---try to prove, the…

A key result in four dimensional black hole physics, since the early 1970s, is Hawking's topology theorem asserting that the cross-sections of an "apparent horizon", separating the black hole region from the rest of the spacetime, are…

We study the relations between the notions of highness, lowness and logical depth in the setting of complexity theory. We introduce a new notion of polylog depth based on time bounded Kolmogorov complexity. We show polylog depth satisfies…

The central conjecture of parameterized complexity states that FPT is not equal to W[1], and is generally regarded as the parameterized counterpart to P != NP. We revisit the issue of the plausibility of FPT != W[1], focusing on two…

We call a continuous path of polygons decreasing if the convex hulls of the polygons form a decreasing family of sets. For an arbitrary polygon of more than three vertices, we characterize the polygons contained in it that can be reached by…

The 15 puzzle is a classic reconfiguration puzzle with fifteen uniquely labeled unit squares within a $4 \times 4$ board in which the goal is to slide the squares (without ever overlapping) into a target configuration. By generalizing the…

Collapse and reverse to collapse explosion transition in self-gravitating systems are studied by molecular dynamics simulations. A microcanonical ensemble of point particles confined to a spherical box is considered; the particles interact…

In a recent article [Phys. Rev. A 94, 052128 (2016)], the authors compute the predictions of two collapse models on the transition probabilities of neutral mesons. Notably, they claim to find an influence on the decay rates and attempt to…

Many classification problems consider classes that form a hierarchy. Classifiers that are aware of this hierarchy may be able to make confident predictions at a coarse level despite being uncertain at the fine-grained level. While it is…

We extend the analysis of the Backgammon model to an ensemble with a fixed number of balls and a fluctuating number of boxes. In this ensemble the model exhibits a first order phase transition analogous to the one in higher dimensional…

A numerical method is proposed to remove the quenched randomness from a data sequence of numbers. A polymer chain of beads is introduced with both a hard core interaction and an appropriate energy associated with the data sequence. The…

Cylindrical algebraic decomposition (CAD) is an important tool for working with polynomial systems, particularly quantifier elimination. However, it has complexity doubly exponential in the number of variables. The base algorithm can be…

The training dynamics of deep neural networks often defy expectations, even as these models form the foundation of modern machine learning. Two prominent examples are grokking, where test performance improves abruptly long after the…

The existence of a finite global attractor for polynomial curve system has been known since the work of Belk et al. [4]. However, except in the hyperbolic case, the rate at which the pullback of a curve under a polynomial converges to the…

Neural collapse describes the geometry of activation in the final layer of a deep neural network when it is trained beyond performance plateaus. Open questions include whether neural collapse leads to better generalization and, if so, why…

Herein, we investigate the zero-error randomized complexity, which is the least cost against the worst input, of AND-OR tree computation by imposing various restrictions on the algorithm to find the Boolean value of the root of that tree…

Low-degree polynomials have emerged as a powerful paradigm for providing evidence of statistical-computational gaps across a variety of high-dimensional statistical models [Wein25]. For detection problems -- where the goal is to test a…

Neural collapse provides an elegant mathematical characterization of learned last layer representations (a.k.a. features) and classifier weights in deep classification models. Such results not only provide insights but also motivate new…

Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms of the constraint language. For…