arXiv research

Proves a conjecture about graph complexes without specific cycle lengths.

Proved inequality for anti-self-polar polytopes.

The paper proves a convex polytope conjecture with specific symmetry conditions.

New cube complexes disprove Kalai's conjecture about sphere facets.

In this note, we present a version of the Thompson sampling algorithm for the problem of online linear generalization with full information (i.e., the experts setting), studied by Kalai and Vempala, 2005. The algorithm uses a Gaussian prior and time-varying Gaussian likelihoods, and we show that it essentially reduces …

The study solves a 2008 problem by Kalai about infinitely many homology-spheres.

We use Klee's Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai's conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel's conjecture that gives an upper bo…

A new approach to group fairness treats it as a bargaining problem.

For $d \geq 2$, Walkup's class ${\cal K}(d)$ consists of the $d$-dimensional simplicial complexes all whose vertex-links are stacked $(d-1)$-spheres. Kalai showed that for $d \geq 4$, all connected members of ${\cal K}(d)$ are obtained from stacked $d$-spheres by finitely many elementary handle additions. According to …

An ongoing aim of research in multiobjective Bayesian optimization is to extend its applicability to a large number of objectives. While coping with a limited budget of evaluations, recovering the set of optimal compromise solutions generally requires numerous observations and is less interpretable since this set tends…

We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL spheres: While the algebraic theory remains the same, we replace the geometric constructions by Pachner's Theorem. This simplifies the reasoning for an important special case of the main result of the first author in arxiv:1812.10454, and alr…

We construct 2^{Ω(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves that asymptotically, there are far more combinatorial types of triangulated 3-spheres …

In recent years, there are many attempts to understand popular heuristics. An example of such a heuristic algorithm is the ID3 algorithm for learning decision trees. This algorithm is commonly used in practice, but there are very few theoretical works studying its behavior. In this paper, we analyze the ID3 algorithm, …

It is known that the $(2k-1)$-sphere has at most $2^{O(n^k \log n)}$ combinatorially distinct triangulations with $n$ vertices, for every $k\ge 2$. Here we construct at least $2^{Ω(n^k)}$ such triangulations, improving on the previous constructions which gave $2^{Ω(n^{k-1})}$ in the general case (Kalai) and $2^{Ω(n^{5/…

Generalized Linear Models (GLMs) and Single Index Models (SIMs) provide powerful generalizations of linear regression, where the target variable is assumed to be a (possibly unknown) 1-dimensional function of a linear predictor. In general, these problems entail non-convex estimation procedures, and, in practice, itera…

This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formu…

In 1987, Kalai proved that stacked spheres of dimension $d\geq 3$ are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension $d=2$. In this article, we give a characterisation of stacked $2$-spheres using what we call the {\em separatio…

New algorithm for reliable learning of Gaussian halfspaces with improved sample and computational complexity.

A constant rebalanced portfolio is an asset allocation algorithm which keeps the same distribution of wealth among a set of assets along a period of time. Recently, there has been work on on-line portfolio selection algorithms which are competitive with the best constant rebalanced portfolio determined in hindsight. By…

Study on Funk geometry volume growth and polytope flags, verifying conjectures.

Algorithm predicts with optimal loss by abstaining from uncertain test examples.

Unified solution to Goodman-Pollack transversal problem using matroids and topology.

Algorithm learns binary function efficiently under arbitrary covariate shift.

Linear optimization is many times algorithmically simpler than non-linear convex optimization. Linear optimization over matroid polytopes, matching polytopes and path polytopes are example of problems for which we have simple and efficient combinatorial algorithms, but whose non-linear convex counterpart is harder and …

FPML algorithm reduces regret by limiting the number of arms chosen per round.

Let $Δ$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A relative lower bound for the number of edges in $Δ$ is that $g_2$ of $Δ$ is at least $g_2$ of the link of any vertex. When this inequality is sharp $Δ$ has relatively minimal $g_2$. For example, whenever the one-skeleton of $Δ$ equals the one-skeleton of …

We give the first dimension-efficient algorithms for learning Rectified Linear Units (ReLUs), which are functions of the form $\mathbf{x} \mapsto \max(0, \mathbf{w} \cdot \mathbf{x})$ with $\mathbf{w} \in \mathbb{S}^{n-1}$. Our algorithm works in the challenging Reliable Agnostic learning model of Kalai, Kanade, and Ma…

New algorithm for online portfolio selection with reduced runtime.

Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's work on polyhedra. Using it and other fundamental combinatorial problems, we intr…