12,899 research outputs found
Non-Redundant Spectral Dimensionality Reduction
Spectral dimensionality reduction algorithms are widely used in numerous
domains, including for recognition, segmentation, tracking and visualization.
However, despite their popularity, these algorithms suffer from a major
limitation known as the "repeated Eigen-directions" phenomenon. That is, many
of the embedding coordinates they produce typically capture the same direction
along the data manifold. This leads to redundant and inefficient
representations that do not reveal the true intrinsic dimensionality of the
data. In this paper, we propose a general method for avoiding redundancy in
spectral algorithms. Our approach relies on replacing the orthogonality
constraints underlying those methods by unpredictability constraints.
Specifically, we require that each embedding coordinate be unpredictable (in
the statistical sense) from all previous ones. We prove that these constraints
necessarily prevent redundancy, and provide a simple technique to incorporate
them into existing methods. As we illustrate on challenging high-dimensional
scenarios, our approach produces significantly more informative and compact
representations, which improve visualization and classification tasks
DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling
This paper explores a fully unsupervised deep learning approach for computing
distance-preserving maps that generate low-dimensional embeddings for a certain
class of manifolds. We use the Siamese configuration to train a neural network
to solve the problem of least squares multidimensional scaling for generating
maps that approximately preserve geodesic distances. By training with only a
few landmarks, we show a significantly improved local and nonlocal
generalization of the isometric mapping as compared to analogous non-parametric
counterparts. Importantly, the combination of a deep-learning framework with a
multidimensional scaling objective enables a numerical analysis of network
architectures to aid in understanding their representation power. This provides
a geometric perspective to the generalizability of deep learning.Comment: 10 pages, 11 Figure
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
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