Related papers: Holographaic Alogorithms on Bases of Rank 2
The theory of holographic algorithms introduced by Valiant represents a novel approach to achieving polynomial-time algorithms for seemingly intractable counting problems via a reduction to counting planar perfect matchings and a linear…
Holographic algorithms with matchgates are a novel approach to design polynomial time computation. It uses Kasteleyn's algorithm for perfect matchings, and more importantly a holographic reduction . The two fundamental parameters of a…
A holographic algorithm solves a problem in domain of size $n$, by reducing it to counting perfect matchings in planar graphs. It may simulate a $n$-value variable by a bunch of $t$ matchgate bits, which has $2^t$ values. The transformation…
The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…
Holographic algorithms introduced by Valiant are composed of two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among…
We provide a recursive description of the signatures realizable on the standard basis by a holographic algorithm. The description allows us to prove tight bounds on the size of planar matchgates and efficiently test for standard signatures.…
Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the…
For any $n\geq 3$ and $ q\geq 3$, we prove that the {\sc Equality} function $(=_n)$ on $n$ variables over a domain of size $q$ cannot be realized by matchgates under holographic transformations. This is a consequence of our theorem on the…
We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances,…
Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate…
We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with $q$ elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus…
Recently holographic prescriptions are proposed to compute quantum complexity of a given state in the boundary theory. A specific proposal known as `holographic subregion complexity' is supposed to calculate the the complexity of a reduced…
We present a random-subspace variant of cubic regularization algorithm that chooses the size of the subspace adaptively, based on the rank of the projected second derivative matrix. Iteratively, our variant only requires access to…
This thesis investigates the central role of homomorphism problems (structure-preserving maps) in two complementary domains: database querying over finite, graph-shaped data, and constraint solving over (potentially infinite) structures.…
We describe a new deep learning architecture for learning to rank question answer pairs. Our approach extends the long short-term memory (LSTM) network with holographic composition to model the relationship between question and answer…
This paper is concerned with the problem of low rank plus sparse matrix decomposition for big data. Conventional algorithms for matrix decomposition use the entire data to extract the low-rank and sparse components, and are based on…
Low-rank plus diagonal (LRPD) decompositions provide a powerful structural model for large covariance matrices, simultaneously capturing global shared factors and localized corrections that arise in covariance estimation, factor analysis,…
In this paper, we propose a domain decomposition dynamical low-rank method to solve high-dimensional radiative transfer problems and similar kinetic equations. The algorithm uses a separate low-rank approximation on each spatial subdomain,…
Holographic Predictive Search (HPS) is a novel approach to search-based hologram generation that uses a mathematical understanding of the optical transforms to make informed optimisation decisions. Existing search techniques such as Direct…
The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan…