251 research outputs found
Communication lower bounds for nested bilinear algorithms
We develop lower bounds on communication in the memory hierarchy or between
processors for nested bilinear algorithms, such as Strassen's algorithm for
matrix multiplication. We build on a previous framework that establishes
communication lower bounds by use of the rank expansion, or the minimum rank of
any fixed size subset of columns of a matrix, for each of the three matrices
encoding the bilinear algorithm. This framework provides lower bounds for any
way of computing a bilinear algorithm, which encompasses a larger space of
algorithms than by fixing a particular dependency graph. Nested bilinear
algorithms include fast recursive algorithms for convolution, matrix
multiplication, and contraction of tensors with symmetry. Two bilinear
algorithms can be nested by taking Kronecker products between their encoding
matrices. Our main result is a lower bound on the rank expansion of a matrix
constructed by a Kronecker product derived from lower bounds on the rank
expansion of the Kronecker product's operands. To prove this bound, we map a
subset of columns from a submatrix to a 2D grid, collapse them into a dense
grid, expand the grid, and use the size of the expanded grid to bound the
number of linearly independent columns of the submatrix. We apply the rank
expansion lower bounds to obtain novel communication lower bounds for nested
Toom-Cook convolution, Strassen's algorithm, and fast algorithms for partially
symmetric contractions.Comment: 37 pages, 5 figures, 1 table. Update includes log-log convex/concave
functions to fix previous bug in v
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of -query quantum algorithms in terms of the
unit ball of a space of degree- polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.Comment: 24 pages, 3 figures. v2: 27 pages, minor changes in response to
referee comment
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC\u2716). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms.
Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms.
We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials
Quantum query algorithms are completely bounded forms
We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC’16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials
Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
The widespread use of multi-sensor technology and the emergence of big
datasets has highlighted the limitations of standard flat-view matrix models
and the necessity to move towards more versatile data analysis tools. We show
that higher-order tensors (i.e., multiway arrays) enable such a fundamental
paradigm shift towards models that are essentially polynomial and whose
uniqueness, unlike the matrix methods, is guaranteed under verymild and natural
conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical
backbone, data analysis techniques using tensor decompositions are shown to
have great flexibility in the choice of constraints that match data properties,
and to find more general latent components in the data than matrix-based
methods. A comprehensive introduction to tensor decompositions is provided from
a signal processing perspective, starting from the algebraic foundations, via
basic Canonical Polyadic and Tucker models, through to advanced cause-effect
and multi-view data analysis schemes. We show that tensor decompositions enable
natural generalizations of some commonly used signal processing paradigms, such
as canonical correlation and subspace techniques, signal separation, linear
regression, feature extraction and classification. We also cover computational
aspects, and point out how ideas from compressed sensing and scientific
computing may be used for addressing the otherwise unmanageable storage and
manipulation problems associated with big datasets. The concepts are supported
by illustrative real world case studies illuminating the benefits of the tensor
framework, as efficient and promising tools for modern signal processing, data
analysis and machine learning applications; these benefits also extend to
vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker
decomposition, HOSVD, tensor networks, Tensor Train
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of t-query quantum algorithms in terms of the unit ball of a
space of degree-2t polynomials. Based on this, we obtain a refined notion of approximate polynomial
degree that equals the quantum query complexity, answering a question of Aaronson et
al. (CCC’16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct.
Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels
to multilinear forms. Using our characterization, we show that many polynomials of degree
four are far from those coming from two-query quantum algorithms. We also give a simple and
short proof of one of the results of Aaronson et al. showing an equivalence between one-query
quantum algorithms and bounded quadratic polynomials
Tensor network representations from the geometry of entangled states
Tensor network states provide successful descriptions of strongly correlated
quantum systems with applications ranging from condensed matter physics to
cosmology. Any family of tensor network states possesses an underlying
entanglement structure given by a graph of maximally entangled states along the
edges that identify the indices of the tensors to be contracted. Recently, more
general tensor networks have been considered, where the maximally entangled
states on edges are replaced by multipartite entangled states on plaquettes.
Both the structure of the underlying graph and the dimensionality of the
entangled states influence the computational cost of contracting these
networks. Using the geometrical properties of entangled states, we provide a
method to construct tensor network representations with smaller effective bond
dimension. We illustrate our method with the resonating valence bond state on
the kagome lattice.Comment: 35 pages, 9 figure
- …