21,973 research outputs found
Nonlinear model order reduction via Dynamic Mode Decomposition
We propose a new technique for obtaining reduced order models for nonlinear
dynamical systems. Specifically, we advocate the use of the recently developed
Dynamic Mode Decomposition (DMD), an equation-free method, to approximate the
nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix
that correlates spatial features while simultaneously associating the activity
with periodic temporal behavior. With this decomposition, one can obtain a
fully reduced dimensional surrogate model and avoid the evaluation of the
nonlinear term in the online stage. This allows for an impressive speed up of
the computational cost, and, at the same time, accurate approximations of the
problem. We present a suite of numerical tests to illustrate our approach and
to show the effectiveness of the method in comparison to existing approaches
Optimality of the Johnson-Lindenstrauss Lemma
For any integers and , we show the existence of a set of vectors such that any embedding satisfying
must have This lower bound matches the upper bound given by the Johnson-Lindenstrauss
lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of
of interest, since there is always an isometric embedding into
dimension (either the identity map, or projection onto
).
Previously such a lower bound was only known to hold against linear maps ,
and not for such a wide range of parameters [LN16]. The
best previously known lower bound for general was [Wel74, Lev83, Alo03], which
is suboptimal for any .Comment: v2: simplified proof, also added reference to Lev8
Bolt: Accelerated Data Mining with Fast Vector Compression
Vectors of data are at the heart of machine learning and data mining.
Recently, vector quantization methods have shown great promise in reducing both
the time and space costs of operating on vectors. We introduce a vector
quantization algorithm that can compress vectors over 12x faster than existing
techniques while also accelerating approximate vector operations such as
distance and dot product computations by up to 10x. Because it can encode over
2GB of vectors per second, it makes vector quantization cheap enough to employ
in many more circumstances. For example, using our technique to compute
approximate dot products in a nested loop can multiply matrices faster than a
state-of-the-art BLAS implementation, even when our algorithm must first
compress the matrices.
In addition to showing the above speedups, we demonstrate that our approach
can accelerate nearest neighbor search and maximum inner product search by over
100x compared to floating point operations and up to 10x compared to other
vector quantization methods. Our approximate Euclidean distance and dot product
computations are not only faster than those of related algorithms with slower
encodings, but also faster than Hamming distance computations, which have
direct hardware support on the tested platforms. We also assess the errors of
our algorithm's approximate distances and dot products, and find that it is
competitive with existing, slower vector quantization algorithms.Comment: Research track paper at KDD 201
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing
and computing with multivariate functions using the functional tensor-train
(FT), a continuous extension of the tensor-train (TT) decomposition. The FT
represents functions using a tensor-train ansatz by replacing the
three-dimensional TT cores with univariate matrix-valued functions. The main
contribution of this paper is a framework to compute the FT that employs
adaptive approximations of univariate fibers, and that is not tied to any
tensorized discretization. The algorithm can be coupled with any univariate
linear or nonlinear approximation procedure. We demonstrate that this approach
can generate multivariate function approximations that are several orders of
magnitude more accurate, for the same cost, than those based on the
conventional approach of compressing the coefficient tensor of a tensor-product
basis. Our approach is in the spirit of other continuous computation packages
such as Chebfun, and yields an algorithm which requires the computation of
"continuous" matrix factorizations such as the LU and QR decompositions of
vector-valued functions. To support these developments, we describe continuous
versions of an approximate maximum-volume cross approximation algorithm and of
a rounding algorithm that re-approximates an FT by one of lower ranks. We
demonstrate that our technique improves accuracy and robustness, compared to TT
and quantics-TT approaches with fixed parameterizations, of high-dimensional
integration, differentiation, and approximation of functions with local
features such as discontinuities and other nonlinearities
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