Recent results have proven the minimax optimality of LASSO and related algorithms for noisy linear regression. However, these results tend to rely on variance estimators that are inefficient or optimizations that are slower than LASSO itself. We propose an efficient estimator for the noise variance in high dimensional linear regression that is significantly faster than LASSO, only requiring p m...

In this paper, we consider stochastic dual coordinate (SDCA) without strongly convex assumption or convex assumption. We show that SDCA converges linearly under mild conditions termed restricted strong convexity. This covers a wide array of popular statistical models including Lasso, group Lasso, and logistic regression with l1 regularization, corrected Lasso and linear regression with SCAD reg...

Stochastic chemical reaction networks constitute a model class to quantitatively describe dynamics and cell-to-cell variability in biological systems. The topology of these networks typically is only partially characterized due to experimental limitations. Current approaches for refining network topology are based on the explicit enumeration of alternative topologies and are therefore restricte...

Methods based on l1-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 l2-error of a Lasso estimate decays at the m...

Given n noisy samples with p dimensions, where n ≪ p, we show that the multi-step thresholding procedure based on the Lasso – we call it the Thresholded Lasso, can accurately estimate a sparse vector β ∈ R in a linear model Y = Xβ + ǫ, where Xn×p is a design matrix normalized to have column l2 norm √ n, and ǫ ∼ N(0, σ2In). We show that under the restricted eigenvalue (RE) condition (Bickel-Rito...

We consider a joint processing of n independent sparse regression problems. Each is based on a sample (yi1, xi1) . . . , (yim, xim) of m i.i.d. observations from yi1 = x T i1βi+εi1, yi1 ∈ R, xi1 ∈ R, i = 1, . . . , n, and εi1 ∼ N(0, σ), say. p is large enough so that the empirical risk minimizer is not consistent. We consider three possible extensions of the lasso estimator to deal with this pr...

We propose kernel block restricted isometry property (KB-RIP) as a generalization of the well-studied RIP and prove a variety of results. First, we present a “sumof-norms”-minimization based formulation of the sparse recovery problem and prove that under suitable conditions on KB-RIP, it recovers the optimal sparse solution exactly. The Group Lasso formulation, widely used as a good heuristic, ...

The additive hazards model has many applications in high-throughput genomic data analysis and clinical studies. In this article, we study the weighted Lasso estimator for the additive hazards model in sparse, high-dimensional settings where the number of time-dependent covariates is much larger than the sample size. Based on compatibility, cone invertibility factors, and restricted eigenvalues ...

The general setting of this work is the constraint-based synthesis of termination arguments. We consider a restricted class of programs called lasso programs. The termination argument for a lasso program is a pair of a ranking function and an invariant. We present the— to the best of our knowledge—first method to synthesize termination arguments for lasso programs that uses linear arithmetic. W...