12,342 research outputs found
Manifold Elastic Net: A Unified Framework for Sparse Dimension Reduction
It is difficult to find the optimal sparse solution of a manifold learning
based dimensionality reduction algorithm. The lasso or the elastic net
penalized manifold learning based dimensionality reduction is not directly a
lasso penalized least square problem and thus the least angle regression (LARS)
(Efron et al. \cite{LARS}), one of the most popular algorithms in sparse
learning, cannot be applied. Therefore, most current approaches take indirect
ways or have strict settings, which can be inconvenient for applications. In
this paper, we proposed the manifold elastic net or MEN for short. MEN
incorporates the merits of both the manifold learning based dimensionality
reduction and the sparse learning based dimensionality reduction. By using a
series of equivalent transformations, we show MEN is equivalent to the lasso
penalized least square problem and thus LARS is adopted to obtain the optimal
sparse solution of MEN. In particular, MEN has the following advantages for
subsequent classification: 1) the local geometry of samples is well preserved
for low dimensional data representation, 2) both the margin maximization and
the classification error minimization are considered for sparse projection
calculation, 3) the projection matrix of MEN improves the parsimony in
computation, 4) the elastic net penalty reduces the over-fitting problem, and
5) the projection matrix of MEN can be interpreted psychologically and
physiologically. Experimental evidence on face recognition over various popular
datasets suggests that MEN is superior to top level dimensionality reduction
algorithms.Comment: 33 pages, 12 figure
Hashing for Similarity Search: A Survey
Similarity search (nearest neighbor search) is a problem of pursuing the data
items whose distances to a query item are the smallest from a large database.
Various methods have been developed to address this problem, and recently a lot
of efforts have been devoted to approximate search. In this paper, we present a
survey on one of the main solutions, hashing, which has been widely studied
since the pioneering work locality sensitive hashing. We divide the hashing
algorithms two main categories: locality sensitive hashing, which designs hash
functions without exploring the data distribution and learning to hash, which
learns hash functions according the data distribution, and review them from
various aspects, including hash function design and distance measure and search
scheme in the hash coding space
Locality Preserving Projections for Grassmann manifold
Learning on Grassmann manifold has become popular in many computer vision
tasks, with the strong capability to extract discriminative information for
imagesets and videos. However, such learning algorithms particularly on
high-dimensional Grassmann manifold always involve with significantly high
computational cost, which seriously limits the applicability of learning on
Grassmann manifold in more wide areas. In this research, we propose an
unsupervised dimensionality reduction algorithm on Grassmann manifold based on
the Locality Preserving Projections (LPP) criterion. LPP is a commonly used
dimensionality reduction algorithm for vector-valued data, aiming to preserve
local structure of data in the dimension-reduced space. The strategy is to
construct a mapping from higher dimensional Grassmann manifold into the one in
a relative low-dimensional with more discriminative capability. The proposed
method can be optimized as a basic eigenvalue problem. The performance of our
proposed method is assessed on several classification and clustering tasks and
the experimental results show its clear advantages over other Grassmann based
algorithms.Comment: Accepted by IJCAI 201
Bilinear Random Projections for Locality-Sensitive Binary Codes
Locality-sensitive hashing (LSH) is a popular data-independent indexing
method for approximate similarity search, where random projections followed by
quantization hash the points from the database so as to ensure that the
probability of collision is much higher for objects that are close to each
other than for those that are far apart. Most of high-dimensional visual
descriptors for images exhibit a natural matrix structure. When visual
descriptors are represented by high-dimensional feature vectors and long binary
codes are assigned, a random projection matrix requires expensive complexities
in both space and time. In this paper we analyze a bilinear random projection
method where feature matrices are transformed to binary codes by two smaller
random projection matrices. We base our theoretical analysis on extending
Raginsky and Lazebnik's result where random Fourier features are composed with
random binary quantizers to form locality sensitive binary codes. To this end,
we answer the following two questions: (1) whether a bilinear random projection
also yields similarity-preserving binary codes; (2) whether a bilinear random
projection yields performance gain or loss, compared to a large linear
projection. Regarding the first question, we present upper and lower bounds on
the expected Hamming distance between binary codes produced by bilinear random
projections. In regards to the second question, we analyze the upper and lower
bounds on covariance between two bits of binary codes, showing that the
correlation between two bits is small. Numerical experiments on MNIST and
Flickr45K datasets confirm the validity of our method.Comment: 11 pages, 23 figures, CVPR-201
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