Sparse Partially Linear Additive Models (2014, background): “…such results (and others of its kind) make potentially strong (and unverifiable) assumptions on the design matrix (e.g., the restricted isometry property, Candes and Tao 2007; the compatibility condition, Bühlmann and van de Geer 2011; van de Geer 2007; small coherence, Candès and Plan 2009, etc.”
Robust Lasso With Missing and Grossly Corrupted Observations (2011, background): “For instance, authors [9], [11] seek to obtain a regression estimate β̂ that delivers small prediction error while others [10], [6] [11] seek to produce a regressor with minimal parameter estimation error, which is measured by the `2-norm of (β̂−β).”
On the conditions used to prove oracle results for the Lasso (2009, background): “The papers [3, 5, 4] show that their coherence conditions imply oracle results and refinements (see also Section 4 for their condition on the diagonal of �). [ 9 ] weaken the coherence conditions by restricting the parameter space for the regression coefficientβ.”
Structured Compressed Sensing: From Theory to Applications (2011, unknown): “It can also be shown that when the distribution used has zero mean and finite variance, then in the asymptotic regime (as M and N grow) the coherence converges to μ(Φ) = 2 √ logN/M [44, 45].”
The Lasso Problem and Uniqueness (2012, background): “For example, see Osborne et al. (2000b), Fuchs (2005), Wainwright (2009), Candes & Plan (2009).”
The Generalized Lasso Problem and Uniqueness (2018, background): “(2013) for an example in the noiseless generalized lasso problem; see Fuchs (2005); Candes and Plan (2009); Wainwright (2009) for examples in the lasso problem; and lastly, see Lee et al.”
The Statistics of Streaming Sparse Regression (2014, background): “A striking difference between Figure 1 and the lasso or least-angle regression paths of Efron et al. [18] is that the lasso path moves along straight lines between knots, whereas our paths look more like Brownian motion once they leave zero.”
A General Theory of Concave Regularization for High-Dimensional Sparse Estimation Problems (2011, unknown): “random β with kβk 0 .n/{kXX⊤/pk 2lnp} and uniformly distributed sgn(β) given kβk 0, θ∗ 1 ≤ 2 and θ∗2 ≤ 1 − 1/ √ 2 with large probability under the incoherence condition maxj6= k |x ⊤ j xk/n| .1/(lnp) [8]. It is worth mentioning that neither the incoherence condition nor the strong irrepresentable co”
Rate Minimaxity of the Lasso and Dantzig Selector for the lq Loss in lr Balls (2010, background): “Candes and Plan (2009) proved the selection consistency of the Lasso under random permutation and sign-change of regression coefficients and a mild condition on the maximum absolute correlation among design vectors.”
Strong rules for discarding predictors in lasso-type problems (2010, background): “For example, Candes and Plan (2009) showed that, if |T | is small, T is drawn from the uniform distribution on |T |-sized subsets of {1, . . . , p}, and each entry of sgn.βÅT / is equal to ±1 with equal probability, then designs X with maxj =k |xTj xk|=O{1= log.p/} satisfy the irrepresentable…”