1,298 research outputs found
Proper Inner Product with Mean Displacement for Gaussian Noise Invariant ICA
Independent Component Analysis (ICA) is a classical method for Blind Source Separation (BSS). In this paper, we are interested in ICA in the presence of noise, i.e., the noisy ICA problem. Pseudo-Euclidean Gradient Iteration (PEGI) is a recent cumulant-based method that defines a pseudo Euclidean inner product to replace a quasi-whitening step in Gaussian noise invariant ICA. However, PEGI has two major limitations: 1) the pseudo Euclidean inner product is improper because it violates the positive definiteness of inner product; 2) the inner product matrix is orthogonal by design but it has gross errors or imperfections due to sample-based estimation. This paper proposes a new cumulant-based ICA method named as PIMD to address these two problems. We first define a Proper Inner product (PI) with proved positive definiteness and then relax the centering preprocessing step to a mean displacement (MD) step. Both PI and MD aim to improve the orthogonality of inner product matrix and the recovery of independent components (ICs) in sample-based estimation. We adopt a gradient iteration step to find the ICs for PIMD. Experiments on both synthetic and real data show the respective effectiveness of PI and MD as well as the superiority of PIMD over competing ICA methods. Moreover, MD can improve the performance of other ICA methods as well
Regression on fixed-rank positive semidefinite matrices: a Riemannian approach
The paper addresses the problem of learning a regression model parameterized
by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear
nature of the search space and on scalability to high-dimensional problems. The
mathematical developments rely on the theory of gradient descent algorithms
adapted to the Riemannian geometry that underlies the set of fixed-rank
positive semidefinite matrices. In contrast with previous contributions in the
literature, no restrictions are imposed on the range space of the learned
matrix. The resulting algorithms maintain a linear complexity in the problem
size and enjoy important invariance properties. We apply the proposed
algorithms to the problem of learning a distance function parameterized by a
positive semidefinite matrix. Good performance is observed on classical
benchmarks
Frame Theory for Signal Processing in Psychoacoustics
This review chapter aims to strengthen the link between frame theory and
signal processing tasks in psychoacoustics. On the one side, the basic concepts
of frame theory are presented and some proofs are provided to explain those
concepts in some detail. The goal is to reveal to hearing scientists how this
mathematical theory could be relevant for their research. In particular, we
focus on frame theory in a filter bank approach, which is probably the most
relevant view-point for audio signal processing. On the other side, basic
psychoacoustic concepts are presented to stimulate mathematicians to apply
their knowledge in this field
Towards music perception by redundancy reduction and unsupervised learning in probabilistic models
PhDThe study of music perception lies at the intersection of several disciplines: perceptual
psychology and cognitive science, musicology, psychoacoustics, and acoustical
signal processing amongst others. Developments in perceptual theory over the last
fifty years have emphasised an approach based on Shannon’s information theory and
its basis in probabilistic systems, and in particular, the idea that perceptual systems
in animals develop through a process of unsupervised learning in response to natural
sensory stimulation, whereby the emerging computational structures are well adapted
to the statistical structure of natural scenes. In turn, these ideas are being applied to
problems in music perception.
This thesis is an investigation of the principle of redundancy reduction through
unsupervised learning, as applied to representations of sound and music.
In the first part, previous work is reviewed, drawing on literature from some of the
fields mentioned above, and an argument presented in support of the idea that perception
in general and music perception in particular can indeed be accommodated within
a framework of unsupervised learning in probabilistic models.
In the second part, two related methods are applied to two different low-level representations.
Firstly, linear redundancy reduction (Independent Component Analysis)
is applied to acoustic waveforms of speech and music. Secondly, the related method of
sparse coding is applied to a spectral representation of polyphonic music, which proves
to be enough both to recognise that the individual notes are the important structural elements,
and to recover a rough transcription of the music.
Finally, the concepts of distance and similarity are considered, drawing in ideas
about noise, phase invariance, and topological maps. Some ecologically and information
theoretically motivated distance measures are suggested, and put in to practice in
a novel method, using multidimensional scaling (MDS), for visualising geometrically
the dependency structure in a distributed representation.Engineering and Physical Science Research Counci
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