26,292 research outputs found
Penalized estimation in large-scale generalized linear array models
Large-scale generalized linear array models (GLAMs) can be challenging to
fit. Computation and storage of its tensor product design matrix can be
impossible due to time and memory constraints, and previously considered design
matrix free algorithms do not scale well with the dimension of the parameter
vector. A new design matrix free algorithm is proposed for computing the
penalized maximum likelihood estimate for GLAMs, which, in particular, handles
nondifferentiable penalty functions. The proposed algorithm is implemented and
available via the R package \verb+glamlasso+. It combines several ideas --
previously considered separately -- to obtain sparse estimates while at the
same time efficiently exploiting the GLAM structure. In this paper the
convergence of the algorithm is treated and the performance of its
implementation is investigated and compared to that of \verb+glmnet+ on
simulated as well as real data. It is shown that the computation time fo
How Correlations Influence Lasso Prediction
We study how correlations in the design matrix influence Lasso prediction.
First, we argue that the higher the correlations are, the smaller the optimal
tuning parameter is. This implies in particular that the standard tuning
parameters, that do not depend on the design matrix, are not favorable.
Furthermore, we argue that Lasso prediction works well for any degree of
correlations if suitable tuning parameters are chosen. We study these two
subjects theoretically as well as with simulations
The degrees of freedom of the Lasso for general design matrix
In this paper, we investigate the degrees of freedom (\dof) of penalized
minimization (also known as the Lasso) for linear regression models.
We give a closed-form expression of the \dof of the Lasso response. Namely,
we show that for any given Lasso regularization parameter and any
observed data belonging to a set of full (Lebesgue) measure, the
cardinality of the support of a particular solution of the Lasso problem is an
unbiased estimator of the degrees of freedom. This is achieved without the need
of uniqueness of the Lasso solution. Thus, our result holds true for both the
underdetermined and the overdetermined case, where the latter was originally
studied in \cite{zou}. We also show, by providing a simple counterexample, that
although the \dof theorem of \cite{zou} is correct, their proof contains a
flaw since their divergence formula holds on a different set of a full measure
than the one that they claim. An effective estimator of the number of degrees
of freedom may have several applications including an objectively guided choice
of the regularization parameter in the Lasso through the \sure framework. Our
theoretical findings are illustrated through several numerical simulations.Comment: A short version appeared in SPARS'11, June 2011 Previously entitled
"The degrees of freedom of penalized l1 minimization
Lossy Compression via Sparse Linear Regression: Computationally Efficient Encoding and Decoding
We propose computationally efficient encoders and decoders for lossy
compression using a Sparse Regression Code. The codebook is defined by a design
matrix and codewords are structured linear combinations of columns of this
matrix. The proposed encoding algorithm sequentially chooses columns of the
design matrix to successively approximate the source sequence. It is shown to
achieve the optimal distortion-rate function for i.i.d Gaussian sources under
the squared-error distortion criterion. For a given rate, the parameters of the
design matrix can be varied to trade off distortion performance with encoding
complexity. An example of such a trade-off as a function of the block length n
is the following. With computational resource (space or time) per source sample
of O((n/\log n)^2), for a fixed distortion-level above the Gaussian
distortion-rate function, the probability of excess distortion decays
exponentially in n. The Sparse Regression Code is robust in the following
sense: for any ergodic source, the proposed encoder achieves the optimal
distortion-rate function of an i.i.d Gaussian source with the same variance.
Simulations show that the encoder has good empirical performance, especially at
low and moderate rates.Comment: 14 pages, to appear in IEEE Transactions on Information Theor
- …