8 research outputs found
A Deterministic Sparse FFT for Functions with Structured Fourier Sparsity
In this paper a deterministic sparse Fourier transform algorithm is presented
which breaks the quadratic-in-sparsity runtime bottleneck for a large class of
periodic functions exhibiting structured frequency support. These functions
include, e.g., the oft-considered set of block frequency sparse functions of
the form as a simple subclass.
Theoretical error bounds in combination with numerical experiments demonstrate
that the newly proposed algorithms are both fast and robust to noise. In
particular, they outperform standard sparse Fourier transforms in the rapid
recovery of block frequency sparse functions of the type above.Comment: 39 pages, 5 figure
A New Class of Fully Discrete Sparse Fourier Transforms: Faster Stable Implementations with Guarantees
In this paper we consider Sparse Fourier Transform (SFT) algorithms for
approximately computing the best -term approximation of the Discrete Fourier
Transform (DFT) of any given input vector
in just -time using only a similarly small number of entries
of . In particular, we present a deterministic SFT algorithm which
is guaranteed to always recover a near best -term approximation of the DFT
of any given input vector in -time. Unlike previous deterministic
results of this kind, our deterministic result holds for both arbitrary vectors
and vector lengths . In addition to these
deterministic SFT results, we also develop several new publicly available
randomized SFT implementations for approximately computing
from using the same general techniques. The best of these new
implementations is shown to outperform existing discrete sparse Fourier
transform methods with respect to both runtime and noise robustness for large
vector lengths
Sparse Fast DCT for Vectors with One-block Support
In this paper we present a new fast and deterministic algorithm for the
inverse discrete cosine transform of type II that reconstructs the input vector
, , with short support of length
from its discrete cosine transform
.
The resulting algorithm has a runtime of and requires
samples of .
In order to derive this algorithm we also develop a new fast and
deterministic inverse FFT algorithm that constructs the input vector
with reflected block support of block length
from with the same runtime and sampling complexities as
our DCT algorithm.Comment: 27 pages, 6 figure
Sparse Harmonic Transforms: A New Class of Sublinear-time Algorithms for Learning Functions of Many Variables
We develop fast and memory efficient numerical methods for learning functions
of many variables that admit sparse representations in terms of general bounded
orthonormal tensor product bases. Such functions appear in many applications
including, e.g., various Uncertainty Quantification(UQ) problems involving the
solution of parametric PDE that are approximately sparse in Chebyshev or
Legendre product bases. We expect that our results provide a starting point for
a new line of research on sublinear-time solution techniques for UQ
applications of the type above which will eventually be able to scale to
significantly higher-dimensional problems than what are currently
computationally feasible.
More concretely, let be a finite Bounded Orthonormal Product Basis (BOPB)
of cardinality . We will develop methods that approximate any function
that is sparse in the BOPB, that is, of the form with
of cardinality . Our method has a runtime of just
, uses only function evaluations on a
fixed and nonadaptive grid, and not more than bits of
memory.
For , the runtime will be less than what is
required to simply enumerate the elements of the basis ; thus our method is
the first approach applicable in a general BOPB framework that falls into the
class referred to as "sublinear-time". This and the similarly reduced sample
and memory requirements set our algorithm apart from previous works based on
standard compressive sensing algorithms such as basis pursuit which typically
store and utilize full intermediate basis representations of size
Inverting Spectrogram Measurements via Aliased Wigner Distribution Deconvolution and Angular Synchronization
We propose a two-step approach for reconstructing a signal from subsampled short-time Fourier transform magnitude
(spectogram) measurements: First, we use an aliased Wigner distribution
deconvolution approach to solve for a portion of the rank-one matrix Second, we use angular
syncrhonization to solve for (and then for
by Fourier inversion). Using this method, we produce two new efficient phase
retrieval algorithms that perform well numerically in comparison to standard
approaches and also prove two theorems, one which guarantees the recovery of
discrete, bandlimited signals from fewer than
STFT magnitude measurements and another which establishes a new class of
deterministic coded diffraction pattern measurements which are guaranteed to
allow efficient and noise robust recovery
(Nearly) Sample-Optimal Sparse Fourier Transform in Any Dimension; RIPless and Filterless
In this paper, we consider the extensively studied problem of computing a
-sparse approximation to the -dimensional Fourier transform of a length
signal. Our algorithm uses samples, is dimension-free,
operates for any universe size, and achieves the strongest
guarantee, while running in a time comparable to the Fast Fourier Transform. In
contrast to previous algorithms which proceed either via the Restricted
Isometry Property or via filter functions, our approach offers a fresh
perspective to the sparse Fourier Transform problem
Lower Memory Oblivious (Tensor) Subspace Embeddings with Fewer Random Bits: Modewise Methods for Least Squares
In this paper new general modewise Johnson-Lindenstrauss (JL) subspace
embeddings are proposed that are both considerably faster to generate and
easier to store than traditional JL embeddings when working with extremely
large vectors and/or tensors.
Corresponding embedding results are then proven for two different types of
low-dimensional (tensor) subspaces. The first of these new subspace embedding
results produces improved space complexity bounds for embeddings of rank-
tensors whose CP decompositions are contained in the span of a fixed (but
unknown) set of rank-one basis tensors. In the traditional vector setting
this first result yields new and very general near-optimal oblivious subspace
embedding constructions that require fewer random bits to generate than
standard JL embeddings when embedding subspaces of spanned by
basis vectors with special Kronecker structure. The second result proven herein
provides new fast JL embeddings of arbitrary -dimensional subspaces
which also require fewer random bits (and so
are easier to store - i.e., require less space) than standard fast JL embedding
methods in order to achieve small -distortions. These new oblivious
subspace embedding results work by effectively folding any given vector
in into a (not necessarily low-rank) tensor, and then
embedding the resulting tensor into for .
Applications related to compression and fast compressed least squares
solution methods are also considered, including those used for fitting low-rank
CP decompositions, and the proposed JL embedding results are shown to work well
numerically in both settings
Sparse Harmonic Transforms II: Best -Term Approximation Guarantees for Bounded Orthonormal Product Bases in Sublinear-Time
In this paper, we develop a sublinear-time compressive sensing algorithm for
approximating functions of many variables which are compressible in a given
Bounded Orthonormal Product Basis (BOPB). The resulting algorithm is shown to
both have an associated best -term recovery guarantee in the given BOPB, and
also to work well numerically for solving sparse approximation problems
involving functions contained in the span of fairly general sets of as many as
orthonormal basis functions. All code is made publicly
available.
As part of the proof of the main recovery guarantee new variants of the well
known CoSaMP algorithm are proposed which can utilize any sufficiently accurate
support identification procedure satisfying a {Support Identification Property
(SIP)} in order to obtain strong sparse approximation guarantees. These new
CoSaMP variants are then proven to have both runtime and recovery error
behavior which are largely determined by the associated runtime and error
behavior of the chosen support identification method. The main theoretical
results of the paper are then shown by developing a sublinear-time support
identification algorithm for general BOPB sets which is robust to arbitrary
additive errors. Using this new support identification method to create a new
CoSaMP variant then results in a new robust sublinear-time compressive sensing
algorithm for BOPB-compressible functions of many variables