7 research outputs found
Direct Exoplanet Detection Using L1 Norm Low-Rank Approximation
We propose to use low-rank matrix approximation using the component-wise
L1-norm for direct imaging of exoplanets. Exoplanet detection by direct imaging
is a challenging task for three main reasons: (1) the host star is several
orders of magnitude brighter than exoplanets, (2) the angular distance between
exoplanets and star is usually very small, and (3) the images are affected by
the noises called speckles that are very similar to the exoplanet signal both
in shape and intensity. We first empirically examine the statistical noise
assumptions of the L1 and L2 models, and then we evaluate the performance of
the proposed L1 low-rank approximation (L1-LRA) algorithm based on visual
comparisons and receiver operating characteristic (ROC) curves. We compare the
results of the L1-LRA with the widely used truncated singular value
decomposition (SVD) based on the L2 norm in two different annuli, one close to
the star and one far away.Comment: 13 pages, 4 figures, BNAIC/BeNeLearn 202
Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations
Low rank matrix approximation is an important tool in machine learning. Given
a data matrix, low rank approximation helps to find factors, patterns and
provides concise representations for the data. Research on low rank
approximation usually focus on real matrices. However, in many applications
data are binary (categorical) rather than continuous. This leads to the problem
of low rank approximation of binary matrix. Here we are given a
binary matrix and a small integer . The goal is to find two binary
matrices and of sizes and respectively, so
that the Frobenius norm of is minimized. There are two models of this
problem, depending on the definition of the dot product of binary vectors: The
model and the Boolean semiring model. Unlike low rank
approximation of real matrix which can be efficiently solved by Singular Value
Decomposition, approximation of binary matrix is -hard even for .
In this paper, we consider the problem of Column Subset Selection (CSS), in
which one low rank matrix must be formed by columns of the data matrix. We
characterize the approximation ratio of CSS for binary matrices. For
model, we show the approximation ratio of CSS is bounded by
and this bound is asymptotically tight. For
Boolean model, it turns out that CSS is no longer sufficient to obtain a bound.
We then develop a Generalized CSS (GCSS) procedure in which the columns of one
low rank matrix are generated from Boolean formulas operating bitwise on
columns of the data matrix. We show the approximation ratio of GCSS is bounded
by , and the exponential dependency on is inherent.Comment: 38 page