We study integer programming instances over polytopes P(A,b)= x:Ax<=b where the constraint matrix A is random, i.e., its entries are i.i.d. Gaussian or, more generally, its rows are i.i.d. from a spherically symmetric distribution. The radius of the largest inscribed ball is closely related to the existence of integer points in the polytope. We show that for m=2^O(sqrt n ), there exist constants c_0 < c_1 such that with high probability, random polytopes are integer feasible if the radius of the largest ball contained in the polytope is at least c_1sqrt log(m n) ; and integer infeasible if the largest ball contained in the polytope is centered at (1 2,...,1 2) and has radius at most c_0sqrt log(m n) . Thus, random polytopes transition from having no integer points to being integer feasible within a constant factor increase in the radius of the largest inscribed ball. We show integer feasibility via a randomized polynomial-time algorithm for finding an integer point in the polytope. Our main tool is a simple new connection between integer feasibility and linear discrepancy. We extend a recent algorithm for finding low-discrepancy solutions (Lovett-Meka, FOCS '12) to give a constructive upper bound on the linear discrepancy of random matrices. By our connection between discrepancy and integer feasibility, this upper bound on linear discrepancy translates to the radius lower bound that guarantees integer feasibility of random polytopes.
Lov ' a sz, Spencer and Vesztergombi @cite_21 showed the following relation between hereditary discrepancy and linear discrepancy.
@cite_21
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Single nucleotide polymorphism (SNP)-heritability estimation is an important topic in several research fields, including animal, plant and human genetics, as well as in ecology. Linear mixed model estimation of SNP-heritability uses the structures of genomic relationships between individuals, which is constructed from genome-wide sets of SNP-markers that are generally weighted equally in their contributions. Proposed methods to handle dependence between SNPs include, “thinning” the marker set by linkage disequilibrium (LD)-pruning, the use of haplotype-tagging of SNPs, and LD-weighting of the SNP-contributions. For improved estimation, we propose a new conceptual framework for genomic relationship matrix, in which Mahalanobis distance-based LD-correction is used in a linear mixed model estimation of SNP-heritability. The superiority of the presented method is illustrated and compared to mixed-model analyses using a VanRaden genomic relationship matrix, a matrix used by GCTA and a matrix employing LD-weighting (as implemented in the LDAK software) in simulated (using real human, rice and cattle genotypes) and real (maize, rice and mice) datasets. Despite of the computational difficulties, our results suggest that by using the proposed method one can improve the accuracy of SNP-heritability estimates in datasets with high LD. This paper addresses the general problem of domain adaptation which arises in a variety of applications where the distribution of the labeled sample available somewhat differs from that of the test data. Building on previous work by Ben- (2007), we introduce a novel distance between distributions, discrepancy distance, that is tailored to adaptation problems with arbitrary loss functions. We give Rademacher complexity bounds for estimating the discrepancy distance from finite samples for different loss functions. Using this distance, we derive novel generalization bounds for domain adaptation for a wide family of loss functions. We also present a series of novel adaptation bounds for large classes of regularization-based algorithms, including support vector machines and kernel ridge regression based on the empirical discrepancy. This motivates our analysis of the problem of minimizing the empirical discrepancy for various loss functions for which we also give novel algorithms. We report the results of preliminary experiments that demonstrate the benefits of our discrepancy minimization algorithms for domain adaptation. This paper addresses the general problem of domain adaptation which arises in a variety of applications where the distribution of the labeled sample available somewhat differs from that of the test data. Building on previous work by Ben- (2007), we introduce a novel distance between distributions, discrepancy distance, that is tailored to adaptation problems with arbitrary loss functions. We give Rademacher complexity bounds for estimating the discrepancy distance from finite samples for different loss functions. Using this distance, we derive new generalization bounds for domain adaptation for a wide family of loss functions. We also present a series of novel adaptation bounds for large classes of regularization-based algorithms, including support vector machines and kernel ridge regression based on the empirical discrepancy. This motivates our analysis of the problem of minimizing the empirical discrepancy for various loss functions for which we also give several algorithms. We report the results of preliminary experiments that demonstrate the benefits of our discrepancy minimization algorithms for domain adaptation. Discrepancy measures how uniformly distributed a point set is with respect to a given set of ranges. Depending on the ranges, several variants arise, including star discrepancy, box discrepancy, and discrepancy of halfspaces. These problems are solvable in time n^O^(^d^), where d is the dimension of the underlying space. As such a dependency on d becomes intractable for high-dimensional data, we ask whether it can be moderated. We answer this question negatively by proving that the canonical decision problems are W[1]-hard with respect to the dimension, implying that no f(d)@?n^O^(^1^)-time algorithm is possible for any function f(d) unless FPT=W[1]. We also discover the W[1]-hardness of other well known problems, such as determining the largest empty box that contains the origin and is inside the unit cube. This is shown to be hard even to approximate within a factor of 2^n.