133 research outputs found
Fast Fourier Transforms for the Rook Monoid
We define the notion of the Fourier transform for the rook monoid (also
called the symmetric inverse semigroup) and provide two efficient
divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing
it. This paper marks the first extension of group FFTs to non-group semigroups
Evaluating prose style transfer with the Bible
In the prose style transfer task a system, provided with text input and a
target prose style, produces output which preserves the meaning of the input
text but alters the style. These systems require parallel data for evaluation
of results and usually make use of parallel data for training. Currently, there
are few publicly available corpora for this task. In this work, we identify a
high-quality source of aligned, stylistically distinct text in different
versions of the Bible. We provide a standardized split, into training,
development and testing data, of the public domain versions in our corpus. This
corpus is highly parallel since many Bible versions are included. Sentences are
aligned due to the presence of chapter and verse numbers within all versions of
the text. In addition to the corpus, we present the results, as measured by the
BLEU and PINC metrics, of several models trained on our data which can serve as
baselines for future research. While we present these data as a style transfer
corpus, we believe that it is of unmatched quality and may be useful for other
natural language tasks as well
Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II
We present a general diagrammatic approach to the construction of efficient
algorithms for computing the Fourier transform of a function on a finite group.
By extending work which connects Bratteli diagrams to the construction of Fast
Fourier Transform algorithms %\cite{sovi}, we make explicit use of the path
algebra connection to the construction of Gel'fand-Tsetlin bases and work in
the setting of quivers. We relate this framework to the construction of a {\em
configuration space} derived from a Bratteli diagram. In this setting the
complexity of an algorithm for computing a Fourier transform reduces to the
calculation of the dimension of the associated configuration space. Our methods
give improved upper bounds for computing the Fourier transform for the general
linear groups over finite fields, the classical Weyl groups, and homogeneous
spaces of finite groups, while also recovering the best known algorithms for
the symmetric group and compact Lie groups.Comment: 53 pages, 5 appendices, 34 figure
Partition Decoupling for Multi-gene Analysis of Gene Expression Profiling Data
We present the extention and application of a new unsupervised statistical
learning technique--the Partition Decoupling Method--to gene expression data.
Because it has the ability to reveal non-linear and non-convex geometries
present in the data, the PDM is an improvement over typical gene expression
analysis algorithms, permitting a multi-gene analysis that can reveal
phenotypic differences even when the individual genes do not exhibit
differential expression. Here, we apply the PDM to publicly-available gene
expression data sets, and demonstrate that we are able to identify cell types
and treatments with higher accuracy than is obtained through other approaches.
By applying it in a pathway-by-pathway fashion, we demonstrate how the PDM may
be used to find sets of mechanistically-related genes that discriminate
phenotypes.Comment: Revise
The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup of a group must be determined from a quantum state over that is uniformly supported on a left coset of . These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups of the -hedral groups, i.e., semidirect products , where , and in particular the affine groups , can be information-theoretically reconstructed using the strong standard method. Moreover, if , these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently
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