arXiv subjects

Hamed Hatami

Publications and source records attributed to Hamed Hatami.

At least 19 recordsLinked to original sources

arXiv2026

A characterization of idempotent Schur multipliers

We prove that every idempotent Schur multiplier is a finite signed sum of contractive idempotent Schur multipliers. This was conjectured by Katavolos and Paulsen in 2003 and previously known only for translation-invariant Schur multipliers, by the Cohen-Host idempotent theorem. Concretely, we show that any boolean matrix $A$ with Schur multiplier norm at most~$\gamma$ (or equivalently $\lVert A\rVert_{\gamma_2} \le \gamma$) can be written as \[ A=\sum_{i=1}^{L}\sigma_i B_i,\] where $L\leq 2^{C\gamma^6}$ for an absolute constant $C$, $\sigma_i\in\{-1,1\}$ are signs, and each $B_i$ is a contractive idempotent Schur multiplier, that is, a boolean matrix whose $1$-entries form a union of all-one rectangular blocks, with no two blocks sharing a row or a column.

math.CA↗

arXiv2026

Communication complexity of point-line incidences over the reals

We construct a point-line incidence problem over the reals whose randomized communication complexity is constant, but whose deterministic communication complexity is linear even when the players have access to an equality oracle. This is the strongest possible separation between these two measures, and it improves on an earlier $O(1)$-versus-$\Omega(\sqrt{n})$ separation of G\"o\"os, Harms, and Riazanov. Because point-line incidence problems have constant sign rank, our construction also bears on a question of Harms and Zamaraev, who asked whether constant sign rank together with constant randomized communication complexity forces constant equality-oracle complexity. This was already refuted by G\"o\"os, Harms, Imbach, and Sokolov with a logarithmic lower bound; our example improves the separation to linear, which is optimal. The proof draws on a construction in the recent disproof of the sum-product conjecture over the reals by Bloom, Sawin, Schildkraut, and Zhelezov, using totally real number fields of large degree and small discriminant.

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arXiv2026

Sign-Rank, Index, and List Replicability: Connections and Separations

In learning theory, the sign-rank of a binary concept class captures the smallest dimension in which it can be represented by points and halfspaces. Despite tremendous interest, lower bounds on sign-rank are notoriously difficult to come by. Two recent approaches to the problem establish lower bounds on sign-rank by measures that are easier to analyze: the $\mathbb{Z}_2$-index and the list replicability number. We order these measures, showing that the $\mathbb{Z}_2$-index is upper-bounded by a linear function of the list replicability number. As a main consequence, we obtain a strong separation between sign-rank and $\mathbb{Z}_2$-index, thereby resolving a question of Frick, Hosseini, and Vasileuski. This motivates a thorough study of list replicability, the stronger of the two lower-bounding measures. We establish upper bounds on the list replicability number by two combinatorial measures: height and minimum star number. We also prove a fundamental composition result, showing that the product of two concept classes has list replicability number bounded by the sum of the list replicability numbers of the two classes.

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arXiv2026

Tight list replicability bounds via a novel sphere covering theorem

In recent years, list replicability has emerged as a framework for formalizing reproducibility in learning theory. A central question is how the required list size relates to the accuracy parameter and natural complexity measures of the hypothesis class. To achieve sharp bounds on list replicability, we prove a novel topological sphere covering theorem, derived from the Borsuk-Ulam theorem. Specifically, if the $d$-sphere is covered by open sets, each of which lies in an open hemisphere, then $d+1$ of these sets must have a common intersection. Using this result, we obtain a sharp bound on the relationship between list size and accuracy for VC classes. We also show that for large-margin half-spaces, provided the margin is not too large, the optimal list size equals the ambient dimension. However, when the margin is taken to be very large, we devise a replicable algorithm achieving the minimal list size of $\lceil d/2 \rceil + 1$.

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arXiv2026

Lower Bounds for Approximate Sign Rank

We prove new upper and lower bounds on $\epsilon$-approximate sign-rank, a relaxation of sign-rank introduced by Chornomaz, Moran, and Waknine (STOC 2025). We show that every $m \times n$ sign matrix with approximate sign-rank $d$ contains a monochromatic rectangle of size $d^{-O(d)}m \times d^{-O(d^2)}n$, paralleling classical results for exact sign-rank. As an application, we establish a lower bound of $\Omega(\sqrt{d/\log d})$ on the $\epsilon$-approximate sign-rank of large-margin $d$-dimensional half-spaces. Prior to our work, the only general lower bound technique known for approximate sign-rank yielded bounds of strength $\epsilon^{-1} - 1$, which are constant for fixed $\epsilon$. A key ingredient is a new geometric theorem on hyperplane avoidance: for any set of $n$ points in general position in $\mathbb{R}^d$, there exist $d$ subsets, each of size $d^{-O(d)} n$, such that no hyperplane simultaneously splits all of them. The proof combines the Forster-Barthe isotropic position theorem with the Bourgain-Tzafriri restricted invertibility principle. We also study the relationship between approximate sign-rank and VC dimension. We prove a lower bound on approximate sign-rank in terms of VC dimension, and exhibit concept classes of VC dimension $2$ with large approximate sign-rank. Finally, we study the approximate sign-rank of the $2^m \times 2^m$ Hadamard matrix $H_m$. The sign-rank of $H_m$ is known to be $\Omega(\sqrt{2^m})$ by Forster's classic theorem. Contrasting this, we adapt an argument of Alman and Williams to show that the approximate sign-rank of $H_m$ is at most $m^{O(\sqrt{m} \log(1/\epsilon))}$, and hence the Hadamard matrix does not witness polynomial-strength lower bounds for approximate sign-rank. Using our VC dimension bound, we prove that the approximate sign-rank of $H_m$ is at least $\Omega_\epsilon(m)$.

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arXiv2025

Simplicial covering dimension of extremal concept classes

Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension. We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.

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arXiv2025

Block structure in boolean matrices of bounded factorization norm

A boolean matrix is blocky if its $1$-entries form a collection of 1-monochromatic submatrices that are disjoint in both rows and columns. Blocky matrices are precisely the set of boolean matrices with $\gamma_2$ factorization norm at most $1$. Building on recent work by Balla, Hambardzumyan, and Tomon, we show that for any boolean matrix with $\gamma_2$ norm at most $\lambda$, there exists a a collection of row- and column-disjoint 1-monochromatic submatrices that together cover a significant portion (at least a $1/2^{2^{O(\lambda)}}$ fraction) of its $1$-entries.

math.CA↗

arXiv2025

Block complexity and idempotent Schur multipliers

We call a matrix blocky if, up to row and column permutations, it can be obtained from an identity matrix by repeatedly applying one of the following operations: duplicating a row, duplicating a column, or adding a zero row or column. Blocky matrices are precisely the boolean matrices that are contractive when considered as Schur multipliers. It is conjectured that any boolean matrix with Schur multiplier norm at most $\gamma$ is expressible as a signed sum \begin{equation*}A = \sum_{i=1}^L \pm B_i\end{equation*} for some blocky matrices $B_i$, where $L$ depends only on $\gamma$. This conjecture is an analogue of Green and Sanders's quantitative version of Cohen's idempotent theorem. In this paper, we prove bounds on $L$ that are polylogarithmic in the dimension of $A$. Concretely, if $A$ is an $n\times n$ matrix, we show that one may take $L = 2^{O(\gamma^7)} \log(n)^2$.

math.CA↗

arXiv2025

Borsuk-Ulam and Replicable Learning of Large-Margin Halfspaces

We prove that the list replicability number of $d$-dimensional $\gamma$-margin half-spaces satisfies \[ \frac{d}{2}+1 \le \mathrm{LR}(H^d_\gamma) \le d, \] which grows with dimension. This resolves several open problems: $\bullet$ Every disambiguation of infinite-dimensional large-margin half-spaces to a total concept class has unbounded Littlestone dimension, answering an open question of Alon, Hanneke, Holzman, and Moran (FOCS '21). $\bullet$ Every disambiguation of the Gap Hamming Distance problem in the large gap regime has unbounded public-coin randomized communication complexity. This answers an open question of Fang, G\"o\"os, Harms, and Hatami (STOC '25). $\bullet$ There is a separation of $O(1)$ vs $\omega(1)$ between randomized and pseudo-deterministic communication complexity. $\bullet$ The maximum list-replicability number of any finite set of points and homogeneous half-spaces in $d$-dimensional Euclidean space is $d$, resolving a problem of Chase, Moran, and Yehudayoff (FOCS '23). $\bullet$ There exists a partial concept class with Littlestone dimension $1$ such that all its disambiguations have infinite Littlestone dimension. This resolves a problem of Cheung, H. Hatami, P. Hatami, and Hosseini (ICALP '23). Our lower bound follows from a topological argument based on a local Borsuk-Ulam theorem. For the upper bound, we construct a list-replicable learning rule using the generalization properties of SVMs.

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arXiv2025

Stability and List-Replicability for Agnostic Learners

Two seminal papers--Alon, Livni, Malliaris, Moran (STOC 2019) and Bun, Livni, and Moran (FOCS 2020)--established the equivalence between online learnability and globally stable PAC learnability in binary classification. However, Chase, Chornomaz, Moran, and Yehudayoff (STOC 2024) recently showed that this equivalence does not hold in the agnostic setting. Specifically, they proved that in the agnostic setting, only finite hypothesis classes are globally stable learnable. Therefore, agnostic global stability is too restrictive to capture interesting hypothesis classes. To address this limitation, Chase et al. introduced two relaxations of agnostic global stability. In this paper, we characterize the classes that are learnable under their proposed relaxed conditions, resolving the two open problems raised in their work. First, we prove that in the setting where the stability parameter can depend on the excess error (the gap between the learner's error and the best achievable error by the hypothesis class), agnostic stability is fully characterized by the Littlestone dimension. Consequently, as in the realizable case, this form of learnability is equivalent to online learnability. As part of the proof of this theorem, we strengthen the celebrated result of Bun et al. by showing that classes with infinite Littlestone dimension are not stably PAC learnable, even if we allow the stability parameter to depend on the excess error. For the second relaxation proposed by Chase et al., we prove that only finite hypothesis classes are globally stable learnable, even if we restrict the agnostic setting to distributions with small population loss.

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arXiv2024

Boolean Functions with Small Approximate Spectral Norm

The sum of the absolute values of the Fourier coefficients of a function $f:\mathbb{F}_2^n \to \mathbb{R}$ is called the spectral norm of $f$. Green and Sanders' quantitative version of Cohen's idempotent theorem states that if the spectral norm of $f:\mathbb{F}_2^n \to \{0,1\}$ is at most $M$, then the support of $f$ belongs to the ring of sets generated by at most $\ell(M)$ cosets, where $\ell(M)$ is a constant that only depends on $M$. We prove that the above statement can be generalized to \emph{approximate} spectral norms if and only if the support of $f$ and its complement satisfy a certain arithmetic connectivity condition. In particular, our theorem provides a new proof of the quantitative Cohen's theorem for $\mathbb{F}_2^n$.

cs.DM↗

arXiv2024

Structure in Communication Complexity and Constant-Cost Complexity Classes

Several theorems and conjectures in communication complexity state or speculate that the complexity of a matrix in a given communication model is controlled by a related analytic or algebraic matrix parameter, e.g., rank, sign-rank, discrepancy, etc. The forward direction is typically easy as the structural implications of small complexity often imply a bound on some matrix parameter. The challenge lies in establishing the reverse direction, which requires understanding the structure of Boolean matrices for which a given matrix parameter is small or large. We will discuss several research directions that align with this overarching theme.

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arXiv2023

Refuting approaches to the log-rank conjecture for XOR functions

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

cs.CC↗

arXiv2023

Sparse graph counting and Kelley-Meka bounds for binary systems

In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without nontrivial three-term arithmetic progressions. In this work, we extend their result, establishing similar bounds for all linear patterns defined by binary systems of linear forms, where "binary" indicates that every linear form depends on exactly two variables. Prior to our work, no strong bounds were known for such systems even in the finite field model setting. A key ingredient in our proof is a graph counting lemma. The classical graph counting lemma, developed by Thomason (Random Graphs 1985) and Chung, Graham, and Wilson (Combinatorica 1989), is a fundamental tool in combinatorics. For a fixed graph $H$, it states that the number of copies of $H$ in a pseudorandom graph $G$ is similar to the number of copies of $H$ in a purely random graph with the same edge density as $G$. However, this lemma is only non-trivial when $G$ is a dense graph. In this work, we prove a graph counting lemma that is also effective when $G$ is sparse. Moreover, our lemma is well-suited for density increment arguments in additive number theory. As an immediate application, we obtain a strong bound for the Tur\'an problem in abelian Cayley sum graphs: let $\Gamma$ be a finite abelian group with odd order. If a Cayley sum graph on $\Gamma$ does not contain any $r$-clique as a subgraph, it must have at most $2^{-\Omega_r(\log^{1/16}|\Gamma|)}\cdot |\Gamma|^2$ edges. These results hinge on the technology developed by Kelley and Meka and the follow-up work by Kelley, Lovett, and Meka (STOC 2024).

math.CO↗

arXiv2023

A tight lower bound on non-adaptive group testing estimation

Efficiently counting or detecting defective items is a crucial task in various fields ranging from biological testing to quality control to streaming algorithms. The \emph{group testing estimation problem} concerns estimating the number of defective elements $d$ in a collection of $n$ total within a given factor. We primarily consider the classical query model, in which a query reveals whether the selected group of elements contains a defective one. We show that any non-adaptive randomized algorithm that estimates the value of $d$ within a constant factor requires $\Omega(\log n)$ queries. This confirms that a known $O(\log n)$ upper bound by Bshouty (2019) is tight and resolves a conjecture by Damaschke and Sheikh Muhammad (2010). Additionally, we prove similar matching upper and lower bounds in the threshold query model.

cs.DS↗

arXiv2023

Online Learning and Disambiguations of Partial Concept Classes

In a recent article, Alon, Hanneke, Holzman, and Moran (FOCS '21) introduced a unifying framework to study the learnability of classes of partial concepts. One of the central questions studied in their work is whether the learnability of a partial concept class is always inherited from the learnability of some ``extension'' of it to a total concept class. They showed this is not the case for PAC learning but left the problem open for the stronger notion of online learnability. We resolve this problem by constructing a class of partial concepts that is online learnable, but no extension of it to a class of total concepts is online learnable (or even PAC learnable).

cs.LG↗

arXiv2022

On depth-3 circuits and covering number: an explicit counter-example

We give a simple construction of $n\times n$ Boolean matrices with $\Omega(n^{4/3})$ zero entries that are free of $2 \times 2$ all-zero submatrices and have covering number $O(\log^4(n))$. This construction provides an explicit counterexample to a conjecture of Pudl\'{a}k, R\"{o}dl and Savick\'{y} and Research Problems 1.33, 4.9, 11.17 of Jukna [Boolean function complexity]. These conjectures were previously refuted by Katz using a probabilistic construction.

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arXiv2021

The Implicit Graph Conjecture is False

An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most $2^{O(n\log(n))}$ graphs on $n$ vertices, and thus has at most factorial speed of growth. The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation. We refute this conjecture by establishing the existence of hereditary graph families with factorial speed of growth that require codes of length $n^{\Omega(1)}$.

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