Geometric Configurations of Algorithms for Reduced m x 2 and 2 x 2 Matrix Multiplication

Matrix multiplication is commonly used in scientific computation. Given matrices A = (aij ) of size l × m and B = (bij ) of size m × n, the standard way to compute the product C := AB is computing cij = Σ^m k=1 aikbkj . In this case, lmn multiplications and ln(m − 1) additions are used. In 1969, V. Strassen found a surprising algorithm to multiply 2 × 2 matrices using 7 multiplications instead of 8 in the standard algorithm. In this way, n × n matrix multiplication can be computed using O(n^log^7 2 ) scalar multiplication operations. If n is large, the Strassen algorithm is much more efficient than the standard algorithm. After Strassen’s algorithm, numerous efforts were made to reduce the complexity for n × n matrix multiplication. By 1986, the bound was reduced to O(n^2.38) by Coppersmith and Winograd. However this is an asymptotic result rather than an implementable algorithm. The complexity has not been significantly improved for 30 years. Matrix multiplication is a tensor and one way to measure the complexity is using its tensor rank. Any tensor can be written as finite sum of rank one tensors and the rank for a tensor is the least number of rank-one tensors needed in the sum. A theorem due to Strassen shows the tensor rank is a good measurement for the complexity. One Bini’s theorem demonstrates that the border rank of the matrix multiplication tensor M is a complexity measurement for matrix multiplication. Even though the problem may sound simple, the border ranks of small matrix multiplication tensors are still unknown. Suppose one wants to compute the border rank of the tensor for the matrix multiplication of size m × 2 and 2 × 2 denoted by R(M). R(M) is closely related to the border rank of reduced matrix multiplication tensor TBCLRS,m, where one entry is set equal to zero. For small m like 2 and 3, there are good geometric configurations in the border rank algorithms for the tensor TBCLRS,m. My project is to understand the geometry of the good existing algorithms in the cases m = 2, 3. In the configuration of case m = 2, the limit 5-plane in the Grassmannian plane in the algorithm intersects with the Segre variety in three special lines. For the case m = 3, the intersection of the limiting 8-plane and the Segre variety consists of the union of a family of lines passing through a plane conic and a special sub-Segre variety. I also try to find analogous algorithms to the m = 2 case or disprove the existence of such algorithms

Algebraic geometry for tensor networks, matrix multiplication, and flag matroids

This thesis is divided into two parts, each part exploring a different topic within the general area of nonlinear algebra. In the first part, we study several applications of tensors. First, we study tensor networks, and more specifically: uniform matrix product states. We use methods from nonlinear algebra and algebraic geometry to answer questions about topology, defining equations, and identifiability of uniform matrix product states. By an interplay of theorems from algebra, geometry, and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch. In addition, we prove a tensor version of the so-called quantum Wielandt inequality, solving an open problem regarding the higher-dimensional version of matrix product states. Second, we present new contributions to the study of fast matrix multiplication. Motivated by the symmetric version of matrix multiplication we study the plethysm S^k(sl_n) of the adjoint representation sl_n of the Lie group SL_n . Moreover, we discuss two algebraic approaches for constructing new tensors which could potentially be used to prove new upper bounds on the complexity of matrix multiplication. One approach is based on the highest weight vectors of the aforementioned plethysm. The other approach uses smoothable finite-dimensional algebras. Finally, we study the Hessian discriminant of a cubic surface, a recently introduced invariant defined in terms of the Waring rank. We express the Hessian discriminant in terms of fundamental invariants. This answers Question 15 of the 27 questions on the cubic surface posed by Bernd Sturmfels. In the second part of this thesis, we apply algebro-geometric methods to study matroids and flag matroids. We review a geometric interpretation of the Tutte polynomial in terms of the equivariant K-theory of the Grassmannian. By generalizing Grassmannians to partial flag varieties, we obtain a new invariant of flag matroids: the flag-geometric Tutte polynomial. We study this invariant in detail, and prove several interesting combinatorial properties

Efficient Algorithms for Graph-Theoretic and Geometric Problems

This thesis studies several different algorithmic problems in graph theory and in geometry. The applications of the problems studied range from circuit design optimization to fast matrix multiplication. First, we study a graph-theoretical model of the so called ''firefighter problem''. The objective is to save as much as possible of an area by appropriately placing firefighters. We provide both new exact algorithms for the case of general graphs as well as approximation algorithms for the case of planar graphs. Next, we study drawing graphs within a given polygon in the plane. We present asymptotically tight upper and lower bounds for this problem Further, we study the problem of Subgraph Isormorphism, which amounts to decide if an input graph (pattern) is isomorphic to a subgraph of another input graph (host graph). We show several new bounds on the time complexity of detecting small pattern graphs. Among other things, we provide a new framework for detection by testing polynomials for non-identity with zero. Finally, we study the problem of partitioning a 3D histogram into a minimum number of 3D boxes and it's applications to efficient computation of matrix products for positive integer matrices. We provide an efficient approximation algorithm for the partitioning problem and several algorithms for integer matrix multiplication. The multiplication algorithms are explicitly or implicitly based on an interpretation of positive integer matrices as 3D histograms and their partitions

Faster Geometric Algorithms via Dynamic Determinant Computation

The computation of determinants or their signs is the core procedure in many important geometric algorithms, such as convex hull, volume and point location. As the dimension of the computation space grows, a higher percentage of the total computation time is consumed by these computations. In this paper we study the sequences of determinants that appear in geometric algorithms. The computation of a single determinant is accelerated by using the information from the previous computations in that sequence. We propose two dynamic determinant algorithms with quadratic arithmetic complexity when employed in convex hull and volume computations, and with linear arithmetic complexity when used in point location problems. We implement the proposed algorithms and perform an extensive experimental analysis. On one hand, our analysis serves as a performance study of state-of-the-art determinant algorithms and implementations. On the other hand, we demonstrate the supremacy of our methods over state-of-the-art implementations of determinant and geometric algorithms. Our experimental results include a 20 and 78 times speed-up in volume and point location computations in dimension 6 and 11 respectively.Comment: 29 pages, 8 figures, 3 table