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— We review the principles of Minimum Description Length and Stochastic Complexity as used in data compression and statistical modeling. Stochastic complexity is formulated as the solution to optimum universal coding problems extending Shannon's basic source coding theorem. The normalized maximized likelihood, mixture, and predictive codings are each shown… (More)

Sparsity or parsimony of statistical models is crucial for their proper interpretations, as in sciences and social sciences. Model selection is a commonly used method to find such models, but usually involves a computationally heavy combinatorial search. Lasso (Tibshirani, 1996) is now being used as a computationally feasible alternative to model selection.… (More)

High-dimensional statistical inference deals with models in which the the number of parameters p is comparable to or larger than the sample size n. Since it is usually impossible to obtain consistent procedures unless p/n → 0, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and… (More)

The first part of this paper proposes an adaptive, data-driven threshold for image denoising via wavelet soft-thresholding. The threshold is derived in a Bayesian framework, and the prior used on the wavelet coefficients is the generalized Gaussian distribution (GGD) widely used in image processing applications. The proposed threshold is simple and… (More)

—Using in situ hyperspectral measurements collected in the Sierra Nevada Mountains in California, we discriminate six species of conifer trees using a recent, nonparametric statistics technique known as penalized discriminant analysis (PDA). A classification accuracy of 76% is obtained. Our emphasis is on providing an intuitive, geometric description of PDA… (More)

The method of wavelet thresholding for removing noise, or denoising, has been researched extensively due to its effectiveness and simplicity. Much of the literature has focused on developing the best uniform threshold or best basis selection. However, not much has been done to make the threshold values adaptive to the spatially changing statistics of… (More)

Consider the standard linear regression model Y = Xβ * +w, where Y ∈ R n is an observation vector, X ∈ R n×d is a design matrix, β * ∈ R d is the unknown regression vector, and w ∼ N (0, σ 2 I) is additive Gaussian noise. This paper studies the minimax rates of convergence for estimation of β * for p-losses and in the 2-prediction loss, assuming that β *… (More)

Methods based on ℓ 1-relaxation, such as basis pursuit and the Lasso, are very popular for sparse regression in high dimensions. The conditions for success of these methods are now well-understood: (1) exact recovery in the noiseless setting is possible if and only if the design matrix X satisfies the restricted nullspace property, and (2) the squared ℓ… (More)

Many statistical machine learning algorithms minimize either an empirical loss function as in Ad-aBoost, or a penalized empirical loss as in Lasso or SVM. A single regularization tuning parameter controls the trade-off between fidelity to the data and generalizability, or equivalently between bias and variance. When this tuning parameter changes, a… (More)