Related papers: The Bit Complexity of Dynamic Algebraic Formulas a…
In the literature of algorithms, the specific computation model is often not explicit as it is assumed that the model of computation is the RAM (Random Access Machine) model. However, the RAM model itself is ill-founded in the literature,…
The correct computation of orbits of discrete dynamical systems on the interval is considered. Therefore, an arbitrary-precision floating-point approach based on automatic error analysis is chosen and a general algorithm is presented. The…
This dissertation focuses on the design and the implementation of domain-specific compilers for linear algebra matrix equations. The development of efficient libraries for such equations, which lie at the heart of most software for…
Linear algebra algorithms often require some sort of iteration or recursion as is illustrated by standard algorithms for Gaussian elimination, matrix inversion, and transitive closure. A key characteristic shared by these algorithms is that…
This paper describes serial and parallel compositional models of multiple objects with part sharing. Objects are built by part-subpart compositions and expressed in terms of a hierarchical dictionary of object parts. These parts are…
Finding the Lie-algebraic closure of a handful of matrices has important applications in quantum computing and quantum control. For most realistic cases, the closure cannot be determined analytically, necessitating an explicit numerical…
We define the complexity of a continuous-time linear system to be the minimum number of bits required to describe its forward increments to a desired level of fidelity, and compute this quantity using the rate distortion function of a…
Many analytics tasks and machine learning problems can be naturally expressed by iterative linear algebra programs. In this paper, we study the incremental view maintenance problem for such complex analytical queries. We develop a…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory…
Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the…
This thesis settles a number of questions related to computational complexity and algebraic, semidefinite programming based relaxations in optimization and control.
We present algorithms for real and complex dot product and matrix multiplication in arbitrary-precision floating-point and ball arithmetic. A low-overhead dot product is implemented on the level of GMP limb arrays; it is about twice as fast…
We investigate the computational complexity of several basic linear algebra primitives, including largest eigenvector computation and linear regression, in the computational model that allows access to the data via a matrix-vector product…
Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate $p$-adic setting to be well-posed. This…
This paper advances the theoretical understanding of active learning label complexity for decision trees as binary classifiers. We make two main contributions. First, we provide the first analysis of the disagreement coefficient for…
We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An $\tilde{O}(n^{\omega+3}a+n^4a^2+n^\omega\log(1/\epsilon))$ time algorithm for finding an…
Every square matrix $A=(a_{uv})\in \mathcal{C}^{n\times n}$ can be represented as a digraph having $n$ vertices. In the digraph, a block (or 2-connected component) is a maximally connected subdigraph that has no cut-vertex. The determinant…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some of the output variables are also input variables, linked by a linear dependency. Fundamental examples…
We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…