Using Elimination Theory to construct Rigid Matrices

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Omega(n). In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in this family are distinct primitive roots of unity of orders roughly exp(n^2 log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we use elimination theory to examine whether the rigidity function is semi-continuous.Comment: 25 Pages, minor typos correcte

Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains

The author, motivated by his results on Hermitian metric rigidity, conjectured in [4] that a proper holomorphic mapping f : Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ≥ 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:= rank(Ω′) ≤ rank(Ω):= r. The Conjecture was resolved in the affirmative by I.-H. Tsai [8]. When the hypothesis r′ ≤ r is removed, the structure of proper holomorphic maps f : Ω → Ω′ is far from being understood, and the complexity in studying such maps depends very much on the difference r′ - r, which is called the rank defect. The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H. Tu [10], in which a rigidity theorem was proven for certain pairs of classical domains of type I, which implies nonexistence theorems for other pairs of such domains. For both results the rank defect is equal to 1, and a generalization of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of (Ω, Ω′) of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and I.-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case. © 2008 Editorial Office of CAM (Fudan University) and Springer-Verlag Berlin Heidelberg.postprin

Static Data Structure Lower Bounds Imply Rigidity

We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of $t \geq \omega(\log^2 n)$ on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space $(s= (1+\varepsilon)n)$, would already imply a semi-explicit ($\bf P^{NP}\rm$) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial ($t\geq n^{\delta}$) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime $(s=n+o(n))$, we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest

Equivalence of Systematic Linear Data Structures and Matrix Rigidity

Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an $NP$ oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between rigidity and the systematic linear model of data structures. For the $n$-dimensional inner product problem with $m$ queries, we prove that lower bounds on the query time imply rigidity lower bounds for the query set itself. In particular, an explicit lower bound of $\omega\left(\frac{n}{r}\log m\right)$ for $r$ redundant storage bits would yield better rigidity parameters than the best bounds due to Alon, Panigrahy, and Yekhanin. We also prove a converse result, showing that rigid matrices directly correspond to hard query sets for the systematic linear model. As an application, we prove that the set of vectors obtained from rank one binary matrices is rigid with parameters matching the known results for explicit sets. This implies that the vector-matrix-vector problem requires query time $\Omega(n^{3/2}/r)$ for redundancy $r \geq \sqrt{n}$ in the systematic linear model, improving a result of Chakraborty, Kamma, and Larsen. Finally, we prove a cell probe lower bound for the vector-matrix-vector problem in the high error regime, improving a result of Chattopadhyay, Kouck\'{y}, Loff, and Mukhopadhyay.Comment: 23 pages, 1 tabl

Large-scale rank and rigidity of the Weil-Petersson metric

We study the large-scale geometry of Weil–Petersson space, that is, Teichmüller space equipped with theWeil–Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space,we recover a result of Eskin,Masur and Rafi which gives the coarse rank of the space. We go on to show that, apart from finitely many cases, the Weil–Petersson spaces are quasi-isometrically distinct, and quasi-isometrically rigid. In particular, any quasi-isometry between such spaces is a bounded distance from an isometry. By a theorem of Brock,Weil–Petersson space is equivariantly quasi-isometric to the pants graph, so our results apply equally well to that space

Properties of Bott manifolds and cohomological rigidity

The cohomological rigidity problem for toric manifolds asks whether the cohomology ring of a toric manifold determines the topological type of the manifold. In this paper, we consider the problem with the class of one-twist Bott manifolds to get an affirmative answer to the problem. We also generalize the result to quasitoric manifolds. In doing so, we show that the twist number of a Bott manifold is well-defined and is equal to the cohomological complexity of the cohomology ring of the manifold. We also show that any cohomology Bott manifold is homeomorphic to a Bott manifold. All these results are also generalized to the case with $\mathbb Z_{(2)}$-coefficients, where $\mathbb Z_{(2)}$ is the localized ring at 2.Comment: 22 page