Related papers: What's Up with Downward Collapse: Using the Easy-H…
Downward collapse (a.k.a. upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial…
Downward translation of equality refers to cases where a collapse of some pair of complexity classes would induce a collapse of some other pair of complexity classes that (a priori) one expects are smaller. Recently, the first downward…
The \emph{strong collapse} of a simplicial complex, proposed by Barmak and Minian (\emph{Disc. Comp. Geom. 2012}), is a combinatorial collapse of a complex onto its sub-complex. Recently, it has received attention from computational…
This article presents a technique for proving problems hard for classes of the polynomial hierarchy or for PSPACE. The rationale of this technique is that some problem restrictions are able to simulate existential or universal quantifiers.…
A problem $\mathcal{P}$ is considered downward self-reducible, if there exists an efficient algorithm for $\mathcal{P}$ that is allowed to make queries to only strictly smaller instances of $\mathcal{P}$. Downward self-reducibility has been…
A decade ago, a beautiful paper by Wagner developed a ``toolkit'' that in certain cases allows one to prove problems hard for parallel access to NP. However, the problems his toolkit applies to most directly are not overly natural. During…
"Period collapse" refers to any situation where the period of the Ehrhart function of a polytope is less than the denominator of that polytope. We study several interesting situations where this occurs, primarily involving triangles. For…
This paper presents the following results on sets that are complete for NP. 1. If there is a problem in NP that requires exponential time at almost all lengths, then every many-one NP-complete set is complete under length-increasing…
In this tutorial, we will survey known results on the complexity of conjunctive query evaluation in different settings, ranging from Boolean queries over counting to more complex models like enumeration and direct access. A particular focus…
Neural Collapse refers to the curious phenomenon in the end of training of a neural network, where feature vectors and classification weights converge to a very simple geometrical arrangement (a simplex). While it has been observed…
We prove a complexity dichotomy theorem for Holant Problems on 3-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted…
We introduce the entangled quantum polynomial hierarchy $\mathsf{QEPH}$ as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove $\mathsf{QEPH}$ collapses to…
Post-training is routinely evaluated through aggregate benchmark scores that treat multi-hop reasoning as a single capability -- as if a model that answers more questions correctly must be better at assembling facts. We show that this…
We describe a procedure called panel collapse for replacing a CAT(0) cube complex $\Psi$ by a "lower complexity" CAT(0) cube complex $\Psi_\bullet$ whenever $\Psi$ contains a codimension-$2$ hyperplane that is extremal in one of the…
This work identifies the existence and cause of a type of posterior collapse that frequently occurs in the Bayesian deep learning practice. For a general linear latent variable model that includes linear variational autoencoders as a…
We study the gravitational collapse of two thin shells of matter, in asymptotically flat spacetime or constrained to move within a spherical box. We show that this simple two-body system has surprisingly rich dynamics, which includes prompt…
In self-supervised representation learning, a common idea behind most of the state-of-the-art approaches is to enforce the robustness of the representations to predefined augmentations. A potential issue of this idea is the existence of…
We introduce a fast and memory efficient approach to compute the persistent homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by…
In this review article, we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked…
A long-standing quantum-mechanical puzzle is whether the collapse of the wave function is a real physical process or simply an epiphenomenon. This puzzle lies at the heart of the measurement problem. One way to choose between the…