20 research outputs found
Local stability and robustness of sparse dictionary learning in the presence of noise
A popular approach within the signal processing and machine learning
communities consists in modelling signals as sparse linear combinations of
atoms selected from a learned dictionary. While this paradigm has led to
numerous empirical successes in various fields ranging from image to audio
processing, there have only been a few theoretical arguments supporting these
evidences. In particular, sparse coding, or sparse dictionary learning, relies
on a non-convex procedure whose local minima have not been fully analyzed yet.
In this paper, we consider a probabilistic model of sparse signals, and show
that, with high probability, sparse coding admits a local minimum around the
reference dictionary generating the signals. Our study takes into account the
case of over-complete dictionaries and noisy signals, thus extending previous
work limited to noiseless settings and/or under-complete dictionaries. The
analysis we conduct is non-asymptotic and makes it possible to understand how
the key quantities of the problem, such as the coherence or the level of noise,
can scale with respect to the dimension of the signals, the number of atoms,
the sparsity and the number of observations
Sparse and spurious: dictionary learning with noise and outliers
A popular approach within the signal processing and machine learning
communities consists in modelling signals as sparse linear combinations of
atoms selected from a learned dictionary. While this paradigm has led to
numerous empirical successes in various fields ranging from image to audio
processing, there have only been a few theoretical arguments supporting these
evidences. In particular, sparse coding, or sparse dictionary learning, relies
on a non-convex procedure whose local minima have not been fully analyzed yet.
In this paper, we consider a probabilistic model of sparse signals, and show
that, with high probability, sparse coding admits a local minimum around the
reference dictionary generating the signals. Our study takes into account the
case of over-complete dictionaries, noisy signals, and possible outliers, thus
extending previous work limited to noiseless settings and/or under-complete
dictionaries. The analysis we conduct is non-asymptotic and makes it possible
to understand how the key quantities of the problem, such as the coherence or
the level of noise, can scale with respect to the dimension of the signals, the
number of atoms, the sparsity and the number of observations.Comment: This is a substantially revised version of a first draft that
appeared as a preprint titled "Local stability and robustness of sparse
dictionary learning in the presence of noise",
http://hal.inria.fr/hal-00737152, IEEE Transactions on Information Theory,
Institute of Electrical and Electronics Engineers (IEEE), 2015, pp.2
Dictionary Identification Results for K-SVD with Sparsity Parameter 1
Publication in the conference proceedings of SampTA, Bremen, Germany, 201
Learning Sparsely Used Overcomplete Dictionaries via Alternating Minimization
We consider the problem of sparse coding, where each sample consists of a
sparse linear combination of a set of dictionary atoms, and the task is to
learn both the dictionary elements and the mixing coefficients. Alternating
minimization is a popular heuristic for sparse coding, where the dictionary and
the coefficients are estimated in alternate steps, keeping the other fixed.
Typically, the coefficients are estimated via minimization, keeping
the dictionary fixed, and the dictionary is estimated through least squares,
keeping the coefficients fixed. In this paper, we establish local linear
convergence for this variant of alternating minimization and establish that the
basin of attraction for the global optimum (corresponding to the true
dictionary and the coefficients) is \order{1/s^2}, where is the sparsity
level in each sample and the dictionary satisfies RIP. Combined with the recent
results of approximate dictionary estimation, this yields provable guarantees
for exact recovery of both the dictionary elements and the coefficients, when
the dictionary elements are incoherent.Comment: Local linear convergence now holds under RIP and also more general
restricted eigenvalue condition