The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following (ordered) matrix removal lemma: For any finite alphabet @math , any hereditary property @math of matrices over @math , and any @math , there exists @math such that for any matrix @math over @math that is @math -far from satisfying @math , most of the @math submatrices of @math do not satisfy @math . Here being @math -far from @math means that one needs to modify at least an @math -fraction of the entries of @math to make it satisfy @math . However, in the above general removal lemma, @math grows very fast as a function of @math , even when @math is characterized by a single forbidden submatrix. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following: For any fixed @math binary matrix @math and any @math there exists @math polynomial in @math , such that for any binary matrix @math in which less than a @math -fraction of the @math submatrices are equal to @math , there exists a set of less than an @math -fraction of the entries of @math that intersects every @math -copy in @math . We generalize the work of Alon, Fischer and Newman [SICOMP'07] and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.
We finish by mentioning several other relevant removal lemma type results. Removal lemmas for vectors (i.e. @ one dimensional matrices where the order is important) are generally easier to obtain; in particular, a removal lemma for vectors over a fixed finite alphabet can be derived from a removal lemma for regular languages proved in @cite_13 . A removal lemma for partially ordered sets with a grid-like structure, which can be seen as a generalization of the removal lemma for vectors, can be deduced from a result of Fischer and Newman in @cite_23 , where they mention that this problem for submatrices is more complicated and not understood. Recently, Ben-Eliezer, Korman and Reichman @cite_19 obtained a removal lemma for patterns in multi-dimensional matrices. A pattern must be taken from locations, whereas in our case the rows and columns of a submatrix need not be consecutive. The case of patterns behaves very differently than that of submatrices, and in particular, in the removal lemma for patterns the parameters are linearly related (for any alphabet size) unlike the case of submatrices (in which, for alphabets of @math letters or more, the relation cannot be polynomial).
@cite_19 @cite_13 @cite_23
2107307520 2555323499 2077575254 2129732816
Binary matrices, and their associated submatrices of 1s, play a central role in the study of random bipartite graphs and in core data mining problems such as frequent itemset mining (FIM). Motivated by these connections, this paper addresses several statistical questions regarding submatrices of 1s in a random binary matrix with independent Bernoulli entries. We establish a three-point concentration result, and a related probability bound, for the size of the largest square submatrix of 1s in a square Bernoulli matrix, and extend these results to non-square matrices and submatrices with fixed aspect ratios. We then consider the noise sensitivity of frequent itemset mining under a simple binary additive noise model, and show that, even at small noise levels, large blocks of 1s leave behind fragments of only logarithmic size. As a result, standard FIM algorithms, which search only for submatrices of 1s, cannot directly recover such blocks when noise is present. On the positive side, we show that an error-tolerant frequent itemset criterion can recover a submatrix of 1s against a background of 0s plus noise, even when the size of the submatrix of 1s is very small. The so-called Berlekamp -- Massey -- Sakata algorithm computes a Grobner basis of a @math -dimensional ideal of relations satisfied by an input table. It extends the Berlekamp -- Massey algorithm to @math -dimensional tables, for @math . We investigate this problem and design several algorithms for computing such a Grobner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations. As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This -like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a matrix (a multivariate generalization of Hankel matrices). Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode @math -cyclic codes, a generalization in dimension @math of Reed Solomon codes. We show that the multi-Hankel matrices are heavily structured when using the ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table. Given an n × m nonnegative matrix A = (a ij ) and positive integral vectors r e ℜ n and c e Âm having a common one-norm h, the (r,c)-scaling problem is to obtain positive diagonal matrices X and Y, if they exist, such that XAY has row and column sums equal to r and c, respectively. The entropy minimization problem corresponding to A is to find an n × m matrix z = (z ij ) having the same zero pattern as A, the sum of whose entries is a given number h, its row and column sums are within given integral vectors of lower and upper bounds, and such that the entropy function consisting of the sum of the terms z ij ln (z ij a ij ) is minimized. When the lower and upper bounds coincide, matrix scaling and entropy minimization are closely related. In this paper we present several complexity bounds for the e-approximate (r,c)-scaling problem, polynomial in n,m,h, 1 e, and ln V v, where V and v are the largest and the smallest positive entries of A, respectively. These bounds, although not polynomial in ln(1 1 e), not only provide alternative complexities for the polynomial time algorithms, but could result in better overall complexities. In particular, our theoretical analysis supports the practicality of the well-known RAS algorithm. In our analysis we obtain bounds on the norm of scaling vectors which will be used in deriving not only some of the above complexities, but also a complexity for square nonnegative matrices having positive permanent. In particular, our results extend, nontrivially, many bounds for the doubly stochastic scaling of square nonnegative matrices previously given by Kalantari and Khachiyan to the case of general (r,c)-scaling. Finally, we study a more general entropy minimization problem where row and column sums are constrained to lie in prescribed intervals, and the sum of all entries is also prescribed. Balinski and Demange described an RAS type algorithm for its continuous version, but did not analyze its complexity. We show that this algorithm produces an 1 e-approximate solution within complexity polynomial in n, m, h, V v and 1 1 e. We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x,y)=f(x)+Q(x,y)+g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to “metrically regular” problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to @math and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with “regular” intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l1 algorithm for compressive sensing and an application to rank reduction problems.