arXiv subjects

Meysam Alishahi

Publications and source records attributed to Meysam Alishahi.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Dimension-Free Sampling for Regularized Classification

We prove optimal sampling bounds achieving $(1\pm\varepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions under various regularization terms. This includes important functions such as logistic and sigmoid loss, hinge loss, and ReLU loss, as prominent and popular representative examples. In particular, we prove $k^2/\varepsilon^2$ upper and lower bounds for $\|\cdot\|_2/k$ regularization, and $k/\varepsilon^2$ upper and lower bounds for $\|\cdot\|_1/k$ regularization. For $\|\cdot\|_2^2/k$ regularization, the sampling complexity depends mainly on a bounded derivative property: if $|g'(x)|\leq g(x)$, and $g(0)>0$, and $g$ is monotonic or convex, then it admits linear in $k$ sampling complexity; otherwise the general bound is $k^2/\varepsilon^2$. However, if $g(0)=0$, our results indicate that no dimension-free bounds are possible, and even sublinear bounds are ruled out. All upper bounds are complemented by matching lower bounds up to polylogarithmic terms. Moreover, our work relies conceptually and algorithmically on simple uniform or (squared) norm sampling and hereby improves over recent cubic $k^3/\varepsilon^2$ sensitivity sampling bounds of (Alishahi and Phillips, ICML'24). This is achieved by refined arguments involving higher moment bounds and empirical process analyses to avoid overcounting that appears in the de-facto standard VC-dimension and sensitivity framework.

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arXiv2026

TabKDE: Simple and Scalable Tabular Data Generation with Kernel Density Estimates

Tabular data generation considers a large table with multiple columns -- each column comprised of numerical, categorical, or sometimes ordinal values. The goal is to produce new rows for the table that replicate the distribution of rows from the original data -- without just copying those initial rows. The last 4 years have seen enormous progress on this problem, mostly using computational expensive methods that employ one-hot encoding, VAEs, and diffusion. This paper describes a new approach to the problem of tabular data generation. By employing copula transformations and modeling the distribution as a kernel density estimate we can nearly match the accuracy and leakage-avoidance achievements of the previous methods, but with almost no training time. Our method is very scalable, and can be run on data sets orders of magnitude larger than prior state-of-the-art on a simple laptop. Moreover, because we employ kernel density estimates, we can store the model as a coreset of the original data -- we believe the first for generative modeling -- and as a result, require significantly less space as well. Our code is available here: \url{https://github.com/tabkde/tabkde-main}

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arXiv2025

The Matching Ramsey Number of Hypergraphs, Revisited

Suppose that a hypergraph ${\mathcal H}$ and an arbitrary nonempty (finite or infinite) set of available colors are given. Each color $x$ is associated with a frequency $τ(x)$, where the set of all such frequencies is bounded. We define a new parameter called the {\it $τ$-matching chromatic number}, denoted by $χ_M(τ, {\mathcal H})$, as the least possible number of colors required to color the edges of ${\mathcal H}$ in such a way that the size of each nonempty monochromatic matching does not exceed the frequency of the corresponding color associated to its edges. The well-known and extensively well-studied chromatic number of general Kneser hypergraph $χ\left( {\rm KG}^r({\mathcal H}) \right)$ is a special case of $χ_M(τ, {\mathcal H})$ when all color frequencies are the fixed constant $r-1$. In this paper, we establish sharp lower bounds for the parameter $χ_M(τ, {\mathcal H})$, utilizing the concepts of the alternation number and the equitable colorability defect.

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arXiv2024

No Dimensional Sampling Coresets for Classification

We refine and generalize what is known about coresets for classification problems via the sensitivity sampling framework. Such coresets seek the smallest possible subsets of input data, so one can optimize a loss function on the coreset and ensure approximation guarantees with respect to the original data. Our analysis provides the first no dimensional coresets, so the size does not depend on the dimension. Moreover, our results are general, apply for distributional input and can use iid samples, so provide sample complexity bounds, and work for a variety of loss functions. A key tool we develop is a Radamacher complexity version of the main sensitivity sampling approach, which can be of independent interest.

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arXiv2023

Linear Distance Metric Learning with Noisy Labels

In linear distance metric learning, we are given data in one Euclidean metric space and the goal is to find an appropriate linear map to another Euclidean metric space which respects certain distance conditions as much as possible. In this paper, we formalize a simple and elegant method which reduces to a general continuous convex loss optimization problem, and for different noise models we derive the corresponding loss functions. We show that even if the data is noisy, the ground truth linear metric can be learned with any precision provided access to enough samples, and we provide a corresponding sample complexity bound. Moreover, we present an effective way to truncate the learned model to a low-rank model that can provably maintain the accuracy in loss function and in parameters -- the first such results of this type. Several experimental observations on synthetic and real data sets support and inform our theoretical results.

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arXiv2019

Topological bounds for graph representations over any field

Haviv ({\em European Journal of Combinatorics}, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $\mathbb{R}$. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over $\mathbb{R}$ -- an important graph invariant from coding theory -- and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed.

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arXiv2019

Maximum Nullity and Forcing Number on Graphs with Maximum Degree at most Three

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $F$ of colored vertices, with all remaining vertices being non-colored. At each time step, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $F$ is called a forcing set of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. The forcing number of a graph $G$, denoted by $F(G)$, is the cardinality of a minimum forcing set of $G$. The maximum nullity of $G$, denoted by $M(G)$, is defined to be the largest possible nullity over all real symmetric matrices $A$ whose $a_{ij} \neq 0$ for $i \neq j$, whenever two vertices $u_{i}$ and $u_{j}$ of $G$ are adjacent. In this paper, we characterize all graphs $G$ of order $n$, maximum degree at most three, and $F(G)=3$. Also we classify these graphs with their maximum nullity.

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arXiv2018

On the random version of the Erdős matching conjecture

The Kneser hypergraph ${\rm KG}^r_{n,k}$ is an $r$-uniform hypergraph with vertex set consisting of all $k$-subsets of $\{1,\ldots,n\}$ and any collection of $r$ vertices forms an edge if their corresponding $k$-sets are pairwise disjoint. The random Kneser hypergraph ${\rm KG}^r_{n,k}(p)$ is a spanning subhypergraph of ${\rm KG}^r_{n,k}$ in which each edge of ${\rm KG}^r_{n,k}$ is retained independently of each other with probability $p$. The independence number of random subgraphs of ${\rm KG}^2_{n,k}$ was recently addressed in a series of works by Bollob{á}s, Narayanan, and Raigorodskii (2016), Balogh, Bollob{á}s, and Narayanan (2015), Das and Tran (2016), and Devlin and Kahn (2016). It was proved that the random counterpart of the Erdős-Ko-Rado theorem continues to be valid even for very small values of $p$. In this paper, generalizing this result, we will investigate the independence number of random Kneser hypergraphs ${\rm KG}^r_{n,k}(p)$. Broadly speaking, when $k$ is much smaller that $n$, we will prove that the random analogue of the Erdős matching conjecture is true even for extremely small values of $p$.

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arXiv2018

A new lower bound for the chromatic number of general Kneser hypergraphs

A general Kneser hypergraph ${\rm KG}^r(\mathcal{H})$ is an $r$-uniform hypergraph that somehow encodes the edge intersections of a ground hypergraph $\mathcal{H}$. The colorability defect of $\mathcal{H}$ is a combinatorial parameter providing a lower bound for the chromatic number of ${\rm KG}^r(\mathcal{H})$ which is addressed in a series of works by Dol'nikov [Sibirskii Matematicheskii Zhurnal, 1988}], Kř\'ıž [Transaction of the American Mathematical Society, 1992], and Ziegler~[Inventiones Mathematicae, 2002]. In this paper, we define a new combinatorial parameter, the equitable colorability defect of hypergraphs, which provides some common improvements of these works. Roughly speaking, we propose a new lower bound for the chromatic number of general Kneser hypergraphs which substantially improves Ziegler's lower bound. It is always as good as Ziegler's lower bound and we provide several families of hypergraphs for which the difference between these two lower bounds is arbitrary large. This specializes to a substantial improvement of the Dol'nikov-Kř\'ıž lower bound for the chromatic number of general Kneser hypergraphs as well. Furthermore, we prove a result ensuring the existence of a colorful subhypergraph in any proper coloring of general Kneser hypergraphs which strengthens Meunier's result [The Electronic Journal of Combinatorics, 2014].

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arXiv2018

Extremal $G$-free induced subgraphs of Kneser graphs

The Kneser graph ${\rm KG}_{n,k}$ is a graph whose vertex set is the family of all $k$-subsets of $[n]$ and two vertices are adjacent if their corresponding subsets are disjoint. The classical Erdős-Ko-Rado theorem determines the cardinality and structure of a maximum induced $K_2$-free subgraph in ${\rm KG}_{n,k}$. As a generalization of the Erdős-Ko-Rado theorem, Erdős proposed a conjecture about the maximum order of an induced $K_{s+1}$-free subgraph of ${\rm KG}_{n,k}$. As the best known result concerning this conjecture, Frankl [Journal of Combinatorial Theory, Series A, 2013], when $n\geq(2s+1)k-s$, gave an affirmative answer to this conjecture and also determined the structure of such a subgraph. In this paper, generalizing the Erdős-Ko-Rado theorem and the Erd{\H o}s matching conjecture, we consider the problem of determining the structure of a maximum family $\mathcal{A}$ for which ${\rm KG}_{n,k}[\mathcal{A}]$ has no subgraph isomorphic to a given graph $G$. In this regard, we determine the size and the structure of such a family provided that $n$ is sufficiently large with respect to $G$ and $k$. Furthermore, for the case $G=K_{1,t}$, we present a Hilton-Milner type theorem regarding above-mentioned problem, which specializes to an improvement of a result by Gerbner et al. [SIAM Journal on Discrete Mathematics, 2012].

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arXiv2017

Fair splitting of colored paths

This paper deals with two problems about splitting fairly a path with colored vertices, where "fairly" means that each part contains almost the same amount of vertices in each color. Our first result states that it is possible to remove one vertex per color from a path with colored vertices so that the remaining vertices can be fairly split into two independent sets of the path. It implies in particular a conjecture of Ron Aharoni and coauthors. The proof uses the octahedral Tucker lemma. Our second result is the proof of a particular case of a conjecture of D{ö}m{ö}t{ö}r P{á}lv{ö}lgyi about fair splittings of necklaces for which one can decide which thieves are advantaged. The proof is based on a rounding technique introduced by Noga Alon and coauthors to prove the discrete splitting necklace theorem from the continuous one.

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arXiv2017

Coloring Properties of Categorical Product of General Kneser Hypergraphs

More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture were generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier (2016) introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful type results for the coloring of categorical product of general Kneser hypergraphs, which generalize the Hajiabolhassan-Meunier result. Also, we present a new lower bound for the chromatic number of categorical product of general Kneser hypergraphs which can be extremely better than the Hajiabolhassan-Meunier lower bound. Using this lower bound, we enrich the family of hypergraphs satisfying Zhu's conjecture.

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arXiv2017

Strengthening topological colorful results for graphs

Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos (Combinatorica, 2006), Simonyi, Tardif, and Zsbán (The Electronic Journal of Combinatorics, 2013), and Chen (Journal of Combinatorial Theory, Series A, 2011). As a consequence of the generalization of Chen's theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisfy Hedetniemi's conjecture for the circular chromatic number.

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arXiv2016

Circular chromatic number of induced subgraphs of Kneser graphs

Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Habolhassan and Zhu [Circular chromatic number of Kneser graphs, Journal of Combinatorial Theory Series B, 2003] proved that if $n$ is sufficiently large with respect to $k$, then the Schrijver graph ${\rm SG}(n,k)$ has the same chromatic and circular chromatic number. Later, Meunier [A topological lower bound for the circular chromatic number of Schrijver graphs, Journal of Graph Theory, 2005] and independently, Simonyi and Tardos [ Local chromatic number, Ky Fan's theorem and circular colorings, Combinatorica, 2006] proved that $χ({\rm SG}(n,k))=χ_c({\rm SG}(n,k))$ if $n$ is even. In this paper, we study the circular chromatic number of induced subgraphs of Kneser graphs. In this regard, we shall first generalize the preceding result to $s$-stable Kneser graphs. Furthermore, as a generalization of Hajiabolhassan and Zhu's result, we prove that if $n$ is large enough with respect to $k$, then any sufficiently large induced subgraph of the Kneser graph ${\rm KG}(n,k)$ has the same chromatic number and circular chromatic number.

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arXiv2016

Chromatic Number of Random Kneser Hypergraphs

Recently, Kupavskii~[{\it On random subgraphs of {K}neser and {S}chrijver graphs. J. Combin. Theory Ser. A, {\rm 2016}.}] investigated the chromatic number of random Kneser graphs $\KG_{n,k}(ρ)$ and proved that, in many cases, the chromatic numbers of the random Kneser graph $\KG_{n,k}(ρ)$ and the Kneser graph $\KG_{n,k}$ are almost surely closed. He also marked the studying of the chromatic number of random Kneser hypergraphs $\KG^r_{n,k}(ρ)$ as a very interesting problem. With the help of $\Z_p$-Tucker lemma, a combinatorial generalization of the Borsuk-Ulam theorem, we generalize Kupavskii's result to random general Kneser hypergraphs by introducing an almost surely lower bound for the chromatic number of them. Roughly speaking, as a special case of our result, we show that the chromatic numbers of the random Kneser hypergraph $\KG^r_{n,k}(ρ)$ and the Kneser hypergraph $\KG^r_{n,k}$ are almost surely closed in many cases. Moreover, restricting to the Kneser and {S}chrijver graphs, we present a purely combinatorial proof for an improvement of Kupavskii's results. Also, for any hypergraph $\HH$, we present a lower bound for the minimum number of colors required in a coloring of $\KG^r(\mathcal{H})$ with no monochromatic $K_{t,\ldots,t}^r$ subhypergraph, where $K_{t,\ldots,t}^r$ is the complete $r$-uniform $r$-partite hypergraph with $t r$ vertices such that each of its parts has $t$ vertices. This result generalizes the lower bound for the chromatic number of $\KG^r(\mathcal{H})$ found by the present authors~[{\it On the chromatic number of general {K}neser hypergraphs. J. Combin. Theory, Ser. B, {\rm 2015}.}].

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arXiv2016

Hedetniemi's Conjecture Via Altermatic Number

A $50$ years unsolved conjecture by Hedetniemi [{\it Homomorphisms of graphs and automata, \newblock {\em Thesis (Ph.D.)--University of Michigan}, 1966}] asserts that the chromatic number of the categorical product of two graphs $G$ and $H$ is $\min\{χ(G),χ(H)\}$. The present authors [{\it On the chromatic number of general {K}neser hypergraphs. \newblock {\em Journal of Combinatorial Theory, Series B}, 2015.}] introduced the altermatic and the strong altermatic number of graphs as two tight lower bounds for the chromatic number of graphs. In this work, we prove a relaxation of Hedetniemi's conjecture in terms of strong altermatic number. Also, we present a tight lower bound for the chromatic number of the categorical product of two graphs in term of their altermatic and strong altermatic numbers. These results enrich the family of pair graphs $\{G,H\}$ satisfying Hedetniemi's conjecture.

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arXiv2016

A Generalization of Gale's lemma

In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two combinatorial sharp lower bounds for ${\rm conid}({\rm B}_0(G))+1$ and ${\rm conid}({\rm B}(G))+2$, two famous topological lower bounds for the chromatic number of a graph $G$.

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arXiv2016

Colorful Subhypergraphs in Uniform Hypergraphs

There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi and Hajiabolhassan [{\it On the chromatic number of general Kneser hypergraphs, Journal of Combinatorial Theory, Series B, 2015}] introduced a lower bound for the chromatic number of Kneser hypergraphs ${\rm KG}^r({\mathcal H})$. Next, Meunier [{\it Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014}] improved their result by proving that any properly colored general Kneser hypergraph ${\rm KG}^r({\mathcal H})$ contains a large colorful $r$-partite subhypergraph provided that $r$ is prime. In this paper, we give some new generalizations of $\mathbb{Z}_p$-Tucker lemma. Hence, improving Meunier's result in some aspects. Some new lower bounds for the chromatic number and local chromatic number of uniform hypergraphs are presented as well.

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