Result Of Kalai Research Articles

Rainbow Fractional Matchings

We prove that any family $E_1, \ldots , E_{\lceil rn \rceil}$ of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$, has a rainbow fractional matching of size $n$ (that is, a set of edges from distinct $E_i$'s which supports such a fractional matching). When the hypergraph is $r$-partite and $n$ is an integer, the number of sets needed goes down from $rn$ to $rn-r+1$. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for $r$-partite hypergraphs with $r>2$. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex set.

The Power of Localization for Efficiently Learning Linear Separators with Noise

We introduce a new approach for designing computationally efficient learning algorithms that are tolerant to noise, and we demonstrate its effectiveness by designing algorithms with improved noise tolerance guarantees for learning linear separators. We consider both the malicious noise model of Valiant [1985] and Kearns and Li [1988] and the adversarial label noise model of Kearns, Schapire, and Sellie [1994]. For malicious noise, where the adversary can corrupt both the label and the features, we provide a polynomial-time algorithm for learning linear separators in ℜ d under isotropic log-concave distributions that can tolerate a nearly information-theoretically optimal noise rate of η = Ω(ϵ), improving on the Ω (ϵ 3 /log 2 ( d/ϵ )) noise-tolerance of Klivans et al. [2009a]. In the case that the distribution is uniform over the unit ball, this improves on the Ω (ϵ/ d 1/4 ) noise-tolerance of Kalai et al. [2005] and the Ω (ϵ 2 /log(d/ϵ)) of Klivans et al. [2009a]. For the adversarial label noise model, where the distribution over the feature vectors is unchanged and the overall probability of a noisy label is constrained to be at most η, we also give a polynomial-time algorithm for learning linear separators in ℜ d under isotropic log-concave distributions that can handle a noise rate of η = Ω(ϵ). In the case of uniform distribution, this improves over the results of Kalai et al. [2005], which either required runtime super-exponential in 1/ϵ (ours is polynomial in 1/ϵ) or tolerated less noise. 1 Our algorithms are also efficient in the active learning setting, where learning algorithms only receive the classifications of examples when they ask for them. We show that, in this model, our algorithms achieve a label complexity whose dependence on the error parameter ϵ is polylogarithmic (and thus exponentially better than that of any passive algorithm). This provides the first polynomial-time active learning algorithm for learning linear separators in the presence of malicious noise or adversarial label noise. Our algorithms and analysis combine several ingredients including aggressive localization, minimization of a progressively rescaled hinge loss, and a novel localized and soft outlier removal procedure. We use localization techniques (previously used for obtaining better sample complexity results) to obtain better noise-tolerant polynomial-time algorithms.

The Cubical Matching Complex

Given a bipartite planar graph embedded in the plane, we define its cubical matching complex. By combining results of Kalai and Propp, we show that the cubical matching complex is collapsible. As a corollary, we obtain that a simply connected region R in the plane that can be tiled with lozenges and hexagons satisfies \({f_0 - f_1 + f_2 - \cdots = 1}\), where fi is the number of tilings with i hexagons. The same relation holds for a region tiled with dominoes and 2 × 2 squares. Furthermore, we show for a region that can be tiled with dominoes, that each link of the associated cubical complex \({\mathcal{C}(R)}\) is either collapsible or homotopy equivalent to a sphere.

Two Decompositions in Topological Combinatorics with Applications to Matroid Complexes

This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on h h -vectors of simplicial polytopes, we show that h h -vectors of pure rank- d d simplicial complexes that have this property satisfy h 0 ≤ h 1 ≤ ⋯ ≤ h [ d / 2 ] h_{0} \leq h_{1} \leq \cdots \leq h_{[d/2]} and h i ≤ h d − i h_{i} \leq h_{d-i} for 0 ≤ i ≤ [ d / 2 ] 0 \leq i \leq [d/2] . We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex ear-decomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the h h -vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [Face numbers inequalities for matroid complexes and Cohen-Macaulay types of Stanley-Reisner rings of distributive lattices, Pacific Journal of Math. 154 (1992), 253-264] that the h h -vector of a matroid complex satisfies the above two sets of inequalities.