Proximal Projection for Doubly Sparse Regularized Models

Gerarda Darlington; Jia Wei He; R. Ayesha Ali

arxiv: 2605.05093 · v1 · submitted 2026-05-06 · 📊 stat.ML · cs.LG· stat.CO· stat.ME

Proximal Projection for Doubly Sparse Regularized Models

Jia Wei He , R. Ayesha Ali , Gerarda Darlington This is my paper

Pith reviewed 2026-05-08 16:24 UTC · model grok-4.3

classification 📊 stat.ML cs.LGstat.COstat.ME

keywords doubly sparse regressionproximal projectionGaussian graphical modelshigh-dimensional regressionlatent variable decompositiongroup regularizationL1-L2 penalty trade-off

0 comments

The pith

Decomposing the coefficient vector into node contributions from a predictor graph allows a proximal projection to enforce double sparsity while trading off L1 and L2 penalties directly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a high-dimensional regression method that assumes predictors form a Gaussian graphical model. It breaks the coefficient vector into latent variables, each capturing one node's total contribution across the graph, then applies regularization to those latents instead of the original coefficients. A custom penalty controls the balance between sparsity types, and optimization uses a novel proximal projection onto the intersection of selected groups rather than duplicating variables. Simulations across varied graphs and node sizes plus real-data examples show the resulting models remain stable compared with other singly or doubly sparse graphical approaches. The work targets settings where computation must be conserved yet key predictors must still be identified.

Core claim

The central claim is that representing the estimated coefficient vector as the sum of latent variables tied to each node in the predictor graph, regularizing those latents with a penalty that explicitly trades L1 against L2 norms, and solving via a proximal projection onto the intersection of the chosen groups produces doubly sparse models that conserve resources relative to duplication methods and exhibit stable performance across different graph structures.

What carries the argument

The proximal projection operator applied to the intersection of selected groups, which directly optimizes the decomposed latent variables without predictor duplication.

If this is right

The method saves computation in high-dimensional settings by avoiding explicit predictor duplication.
A single penalty parameter directly controls the L1-L2 trade-off on the latent node contributions.
Stable recovery of relevant predictors holds across different graph topologies and node counts.
The approach generates models that are both sparse in the coefficient values and in the selected nodes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same decomposition could be tested on networks that are not Gaussian, such as those arising in genomics or social data.
If the graph must be estimated rather than given, the overall procedure would need an outer loop whose error propagation is not yet quantified.
The projection step may scale to very large node sets more readily than duplication-based alternatives.

Load-bearing premise

The predictors can be represented by a Gaussian graphical model whose structure is known or estimable and can be exploited for the decomposition.

What would settle it

A simulation in which the assumed graph is misspecified and the method's prediction error or stability drops below that of a standard singly sparse graphical lasso.

Figures

Figures reproduced from arXiv: 2605.05093 by Gerarda Darlington, Jia Wei He, R. Ayesha Ali.

**Figure 1.** Figure 1: Undirected consensus graph for the blood brain barrier data view at source ↗

**Figure 2.** Figure 2: MSE for 90 permutations of blood-brain data view at source ↗

**Figure 3.** Figure 3: MSE for 90 permutations of ADNI data view at source ↗

read the original abstract

Regularization is often used in high-dimensional regression settings to generate a sparse model, which can save tremendous computing resources and identify predictors that are most strongly associated with the response. When the predictors can be represented by a Gaussian graphical model, the structure of the predictor graph can be exploited during regularization. Our proposed model exploits this underlying predictor graph structure by decomposing the estimated coefficient vector into a sum of latent variables that correspond to the sum of each node contribution to the coefficient vector. Regularization is then performed on the latent variables rather than on the coefficient vector directly. We use a penalty function that permits a clear user-defined trade-off between the L1 and L2 penalties and propose a novel proximal projection during optimization. Further, our implementation computes the projection operator for the intersection of selected groups, which conserves more computing resources compared to predictor duplication methods, especially for high-dimensional data. Through simulation, we evaluate the performance of our approach under different graph structures and node counts, and present results on real-world data. Results suggest that our method exhibits stable performance relative to other singly or doubly sparse graphical regression models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's main addition is an efficient proximal projection operator for group intersections in doubly sparse regression that exploits a known predictor graph, but real-data claims rest on how that graph is obtained.

read the letter

The core contribution is a decomposition of the coefficient vector into latent node contributions tied to the predictor Gaussian graphical model, followed by a mixed L1/L2 penalty optimized via a proximal projection on the intersection of selected groups. This avoids the predictor-duplication trick and should cut compute in high dimensions. The implementation detail for the projection operator looks like the genuinely new piece not covered in the cited prior work on singly or doubly sparse models. Simulations under varied graph structures and node counts show stable performance against baselines, which is a reasonable empirical check for the method as described. The real-data section is mentioned but not detailed enough in the abstract to judge effect sizes or controls. The soft spot is exactly the one the stress-test flags: everything depends on accurate knowledge of the predictor graph. Simulations give the graph for free, but the abstract gives no indication of how the graph is recovered or estimated for real data, nor any robustness checks under graph error or misspecification. If the graph step is noisy, the latent decomposition may not deliver the claimed stability. This is for people already working on high-dimensional regression with graphical covariate structure who need a faster projection step. It is not broad enough to reshape the field, but the optimization angle is concrete enough that a serious referee should see it, with the expectation that revisions will clarify the real-data graph handling and add sensitivity checks.

Referee Report

2 major / 1 minor

Summary. The paper proposes a proximal projection method for doubly sparse regularized regression that exploits a Gaussian graphical model structure on the predictors. The coefficient vector is decomposed into latent variables representing node contributions, a mixed L1/L2 penalty is applied, and a novel proximal projection operator is used on the intersection of selected groups to achieve efficiency over duplication-based approaches. Simulations under varied graph structures and node counts, plus real data experiments, are used to claim stable performance relative to other singly or doubly sparse graphical regression models.

Significance. If the empirical claims hold under realistic conditions, the method could provide a practical, tunable way to incorporate known or estimated predictor graph structure into high-dimensional regression while conserving computation. The proximal projection for group intersections is a potentially useful technical contribution for optimization in structured penalties.

major comments (2)

[Abstract and real data results] The central claim of stable performance on real data (Abstract) is load-bearing for the paper's contribution but rests on the unverified assumption that the predictor graph is accurately known or recovered. The manuscript provides no description of how the graph is obtained or estimated for the real data application, nor any experiments assessing sensitivity to graph misspecification or estimation error, unlike the simulations which assume known structures.
[Introduction and simulation setup] The weakest assumption identified in the approach—that predictors follow a Gaussian graphical model allowing exploitation of the graph—is not stress-tested for robustness. Without such analysis, the performance advantage over baselines cannot be confidently attributed to the proximal projection method rather than perfect graph knowledge.

minor comments (1)

[Methods] Notation for the latent variable decomposition and the mixed penalty function could be clarified with an explicit example or small-scale illustration to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments below and will incorporate revisions to clarify the real-data graph construction and to add robustness checks for graph misspecification.

read point-by-point responses

Referee: [Abstract and real data results] The central claim of stable performance on real data (Abstract) is load-bearing for the paper's contribution but rests on the unverified assumption that the predictor graph is accurately known or recovered. The manuscript provides no description of how the graph is obtained or estimated for the real data application, nor any experiments assessing sensitivity to graph misspecification or estimation error, unlike the simulations which assume known structures.

Authors: We agree that the manuscript lacks an explicit description of how the predictor graph was obtained or estimated in the real-data experiments and does not include sensitivity analysis to graph misspecification. In the revised version we will add a dedicated paragraph detailing the graph estimation procedure applied to the real data and will include new simulation results that introduce controlled graph estimation error to assess whether the reported performance advantages persist under imperfect graph information. revision: yes
Referee: [Introduction and simulation setup] The weakest assumption identified in the approach—that predictors follow a Gaussian graphical model allowing exploitation of the graph—is not stress-tested for robustness. Without such analysis, the performance advantage over baselines cannot be confidently attributed to the proximal projection method rather than perfect graph knowledge.

Authors: The current simulations assume a known graph, which is the natural setting for evaluating a method that exploits graph structure. We acknowledge that this leaves open the question of robustness to graph error. We will revise the introduction to state this assumption more explicitly and will add a new subsection in the simulation studies that systematically varies the accuracy of the supplied graph (via edge perturbations and edge omission) to isolate the contribution of the proximal projection operator from perfect graph knowledge. revision: yes

Circularity Check

0 steps flagged

No circularity in the proposed optimization derivation

full rationale

The paper proposes a decomposition of the coefficient vector into latent node contributions based on an assumed Gaussian graphical model for predictors, followed by a mixed L1/L2 penalty and a novel proximal projection operator for group intersections. These elements are presented as direct constructions from optimization principles and penalty design, with no equations or steps shown to reduce by construction to fitted parameters, self-definitions, or prior self-citations. Simulations under varied graphs and real-data results function as external empirical checks rather than tautological predictions. No load-bearing self-citation chains, uniqueness theorems, or ansatz smuggling are identifiable from the abstract or claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of Gaussian graphical model structure for predictors; no free parameters or new entities introduced in the abstract description.

axioms (1)

domain assumption Predictors can be represented by a Gaussian graphical model
Explicitly stated in the abstract as the basis for exploiting graph structure during regularization.

pith-pipeline@v0.9.0 · 5500 in / 1061 out tokens · 37732 ms · 2026-05-08T16:24:26.861040+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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mini-mental state

Ando, T. & Bai, J. (2017), ‘Clustering huge number of financial time series: A panel data approach with high-dimensional predictors and factor structures’, Journal of the American Statistical Association 112(519), 1182–1198. Bauschke, H. & Combettes, P. (2011), Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer-Verlag. Beck, A. & Teb...

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URL: https://doi.org/10.1038/s41586-020-2649-2 Hartig, M., Truran-Sacrey, D., Raptentsetsang, S., Simonson, A., Mezher, A., Schuff, N., Weiner, M. et al. (2014), ‘Ucsf freesurfer methods’, ADNI Alzheimers Disease Neuroimag- ing Initiative: San Francisco, CA, USA

work page doi:10.1038/s41586-020-2649-2 2014
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Hoerl, A. E. & Kennard, R. W. (1970), ‘Ridge regression: Biased estimation for nonorthog- onal problems’, Technometrics 12(1), 55–67. Hu, C., Cheng, L., Sepulcre, J., Johnson, K. A., Fakhri, G. E., Lu, Y. M. & Li, Q. (2015), ‘A spectral graph regression model for learning brain connectivity of Alzheimer’s disease’, PloS one 10(5), e0128136. Jack Jr, C. R....

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(2015), ‘Caret: classification and regression training’, Astrophysics Source Code Library pp

Kuhn, M. (2015), ‘Caret: classification and regression training’, Astrophysics Source Code Library pp. ascl–1505. Lauritzen, S. L. (1996), Graphical models, Vol. 17, Clarendon Press. Leeuwis, A. E., Benedictus, M. R., Kuijer, J. P., Binnewijzend, M. A., Hooghiemstra, A. M., Verfaillie, S. C., Koene, T., Scheltens, P., Barkhof, F., Prins, N. D. et al. (201...

work page arXiv 2015
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& Tibshirani, R

Simon, N., Friedman, J., Hastie, T. & Tibshirani, R. (2013), ‘A sparse-group lasso’, Journal of computational and graphical statistics 22(2), 231–245. Stephenson, M., Darlington, G. A., Schenkel, F. S., Squires, E. J. & Ali, R. A. (2019), ‘DSRIG: incorporating graphical structure in the regularized modeling of SNP data’, Journal of Bioinformatics and Comp...

work page 2013
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Zhang, D., Shen, D., Initiative, A. D. N. et al. (2012), ‘Multi-modal multi-task learning for joint prediction of multiple regression and classification variables in Alzheimer’s disease’, NeuroImage 59(2), 895–907. Zou, H. (2006), ‘The adaptive lasso and its oracle properties’, Journal of the American statistical association 101(476), 1418–1429. 36 A Appe...

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[7]

Lemma 3 can be proved by verifying the properties of a norm

The regularizer ΩG p is a norm. Lemma 3 can be proved by verifying the properties of a norm. A regularizer with double sparsity ( l1 and l2) has been proven to be a norm by Stephenson et al. (2019), Rao et al. (2013). We denote two subspaces M ⊆ M of Rp and its orthogonal complement M ⊥ to ensure the decomposability of the regularizer, where M ⊥ is the se...

work page 2019
[8]

Then the norm R(β) in the SGLIG model is decomposable with respect to the subspace pair MSG , M ⊥ SG

Assume (A1)-(A3). Then the norm R(β) in the SGLIG model is decomposable with respect to the subspace pair MSG , M ⊥ SG. It is straightforward to prove lemma 4, since the components in MSG and M ⊥ SG are clearly non-overlapping under assumption A2. Given R(β) is a decomposable norm, we can show that the dual norm of R(β) is upper bounded. Assumption A5 ass...

work page 2012
[9]

The definition summarized by Negahban et al

The loss function L(β) will satisfy a Restricted Strong Convexity (RSC) condition with curvature parameter κL if it is convex, differentiable, and δL(∆, β) ≥ κL∥∆∥2 2, ∀∆ ∈ C(MSG , M ⊥ SG, β), where ⟨·, ·⟩ represents the inner product of two vectors, and C(·) is as defined in Equation (3.8). The definition summarized by Negahban et al. (2012) includes a t...

work page 2012
[10]

Denote u as a p-dimensional vector and uNi such that (uNi)j ̸= 0 for j ∈ N i

Assume A4. Denote u as a p-dimensional vector and uNi such that (uNi)j ̸= 0 for j ∈ N i. Then the dual norm of R(β) in the SGLIG model is upper bounded as follows: R∗(u) ≤ max i=1,...,p 1√ dmin ∥uG∥2. Proof. R∗(u) = max β {βT u} s.t. R(β) ≤ 1 = max V ( pX i=1 V (i)T uNi ) s.t. pX i=1 ατi∥V (i)∥2 + (1 − α) p di∥V (i)∥1 ≤ 1 ≤ max V ( pX i=1 V (i)T uNi ) s.t...

work page 2012
[11]

(2019)) For chi-square random vari- ables X1, X2,

(Proposition S.5 Stephenson et al. (2019)) For chi-square random vari- ables X1, X2, . . . , Xp with d degrees of freedom and some constant c, P max i=1,...,p Xi ≤ c2d ≥ 1 − exp log(p) − (c − 1)2d 2 . Lemma

work page 2019
[12]

Then the upper bound of R (∇L)2 is presented by setting c = r√n and r2 = log(p)+dmax dmax

Then by Lemma 7, P max i=1,...,p Vi ≤ c2d ≥ 1 − exp log(p) − (c − 1)2d 2 followed with R∗(∇L)2 ≤ σ∗σ2 n2dmin max i=1,...,p ∥Vi∥2 2 ≤ σ∗σ2 n2dmin c2dmax n with probability ≥ 1 − exp log(p) − (c − 1)2dmax 2 . Then the upper bound of R (∇L)2 is presented by setting c = r√n and r2 = log(p)+dmax dmax . Stephenson et al. (2019) provide explicit derivation of th...

work page 2019
[13]

Assume (A1)-(A3), and let ΠMSG (·) represent the projection onto the subspace MSG

with a strictly positive regularization parameter satisfying λ ≥ 2R∗(∇L(β)), where R∗(·) is the dual norm of R(·) and ∇L(β) is the gradient of the loss function. Assume (A1)-(A3), and let ΠMSG (·) represent the projection onto the subspace MSG . Then, the error, ˆ∆ = ˆβ − β, will belong to the set: C(MSG , M ⊥ SG, β) := n ∆ ∈ Rp | R(ΠM⊥ SG ∆) ≤ 3R[ΠMSG (∆...

work page 2012
[14]

Hence, we can summarize that : Qn(u) − Qn(0) = I1 + I2 + I3, where I1 d → 1 2 uT M u − uT W, while I3 + I4 d → 0 if supp(u) ⊆ J0 and I3 + I4 d → ∞ otherwise

Then we have |I4| → ∞ as n → ∞. Hence, we can summarize that : Qn(u) − Qn(0) = I1 + I2 + I3, where I1 d → 1 2 uT M u − uT W, while I3 + I4 d → 0 if supp(u) ⊆ J0 and I3 + I4 d → ∞ otherwise. Therefore, we can conclude that Qn(u) − Qn(0) d → D(u), where D(u) = 8 >>>< >>>: 1 2 uT M u − uT W, if supp (u) ⊆ J0, ∞, otherwise. Since ˆu = arg minu∈Rp Qn(u) = arg ...

work page 1996
[15]

48 A.5 Comparisons of Finite Error Bound Between the SGLIG and DSRIG Scenario 1: Comparisons Under Assumptions of DSRIG The DSRIG uses the adaptive weights for the l2 penalty defined as τi = √di cov(X,Yi) , where |cov(X, Yi) ≤ 1| for ∀i ∈ {1, . . . , p}. Therefore, we have τ 2 i ≥ √di for ∀i ∈ {1, . . . , p}. Then it can be shown that τ 2 min ≥ dmin and 1...

work page 2019
[16]

, p in assumption A4., which means τ 2 min ≥ dmin

The SGLIG model assumes τi is lower bounded by √di for i = 1 , . . . , p in assumption A4., which means τ 2 min ≥ dmin. However, The SGLIG model uses different adaptive weights for the l2 norm which is defined as τi = 1 cov(X,Yi) for ∀i ∈ { 1, . . . , p}, so there is no direct scale condition between τmin and √dmin when fitting the SGLIG 49 model to the s...

work page 2003
[17]

(2019) used 135 volume measurements ( mm3) obtained from the Freesurfer segmented data as the predictors

and Stephenson et al. (2019) used 135 volume measurements ( mm3) obtained from the Freesurfer segmented data as the predictors. FreeSurfer is an image analysis suite used to process the MRI data collected in ADNI. It was developed by Hartig et al. (2014) and is available online (http://surfer.nmr.mgh.harvard.edu/). We are also interested in predicting the...

work page 2019

[1] [1]

mini-mental state

Ando, T. & Bai, J. (2017), ‘Clustering huge number of financial time series: A panel data approach with high-dimensional predictors and factor structures’, Journal of the American Statistical Association 112(519), 1182–1198. Bauschke, H. & Combettes, P. (2011), Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer-Verlag. Beck, A. & Teb...

work page 2017

[2] [2]

URL: https://doi.org/10.1038/s41586-020-2649-2 Hartig, M., Truran-Sacrey, D., Raptentsetsang, S., Simonson, A., Mezher, A., Schuff, N., Weiner, M. et al. (2014), ‘Ucsf freesurfer methods’, ADNI Alzheimers Disease Neuroimag- ing Initiative: San Francisco, CA, USA

work page doi:10.1038/s41586-020-2649-2 2014

[3] [3]

Hoerl, A. E. & Kennard, R. W. (1970), ‘Ridge regression: Biased estimation for nonorthog- onal problems’, Technometrics 12(1), 55–67. Hu, C., Cheng, L., Sepulcre, J., Johnson, K. A., Fakhri, G. E., Lu, Y. M. & Li, Q. (2015), ‘A spectral graph regression model for learning brain connectivity of Alzheimer’s disease’, PloS one 10(5), e0128136. Jack Jr, C. R....

work page 1970

[4] [4]

(2015), ‘Caret: classification and regression training’, Astrophysics Source Code Library pp

Kuhn, M. (2015), ‘Caret: classification and regression training’, Astrophysics Source Code Library pp. ascl–1505. Lauritzen, S. L. (1996), Graphical models, Vol. 17, Clarendon Press. Leeuwis, A. E., Benedictus, M. R., Kuijer, J. P., Binnewijzend, M. A., Hooghiemstra, A. M., Verfaillie, S. C., Koene, T., Scheltens, P., Barkhof, F., Prins, N. D. et al. (201...

work page arXiv 2015

[5] [5]

& Tibshirani, R

Simon, N., Friedman, J., Hastie, T. & Tibshirani, R. (2013), ‘A sparse-group lasso’, Journal of computational and graphical statistics 22(2), 231–245. Stephenson, M., Darlington, G. A., Schenkel, F. S., Squires, E. J. & Ali, R. A. (2019), ‘DSRIG: incorporating graphical structure in the regularized modeling of SNP data’, Journal of Bioinformatics and Comp...

work page 2013

[6] [6]

Zhang, D., Shen, D., Initiative, A. D. N. et al. (2012), ‘Multi-modal multi-task learning for joint prediction of multiple regression and classification variables in Alzheimer’s disease’, NeuroImage 59(2), 895–907. Zou, H. (2006), ‘The adaptive lasso and its oracle properties’, Journal of the American statistical association 101(476), 1418–1429. 36 A Appe...

work page 2012

[7] [7]

Lemma 3 can be proved by verifying the properties of a norm

The regularizer ΩG p is a norm. Lemma 3 can be proved by verifying the properties of a norm. A regularizer with double sparsity ( l1 and l2) has been proven to be a norm by Stephenson et al. (2019), Rao et al. (2013). We denote two subspaces M ⊆ M of Rp and its orthogonal complement M ⊥ to ensure the decomposability of the regularizer, where M ⊥ is the se...

work page 2019

[8] [8]

Then the norm R(β) in the SGLIG model is decomposable with respect to the subspace pair MSG , M ⊥ SG

Assume (A1)-(A3). Then the norm R(β) in the SGLIG model is decomposable with respect to the subspace pair MSG , M ⊥ SG. It is straightforward to prove lemma 4, since the components in MSG and M ⊥ SG are clearly non-overlapping under assumption A2. Given R(β) is a decomposable norm, we can show that the dual norm of R(β) is upper bounded. Assumption A5 ass...

work page 2012

[9] [9]

The definition summarized by Negahban et al

The loss function L(β) will satisfy a Restricted Strong Convexity (RSC) condition with curvature parameter κL if it is convex, differentiable, and δL(∆, β) ≥ κL∥∆∥2 2, ∀∆ ∈ C(MSG , M ⊥ SG, β), where ⟨·, ·⟩ represents the inner product of two vectors, and C(·) is as defined in Equation (3.8). The definition summarized by Negahban et al. (2012) includes a t...

work page 2012

[10] [10]

Denote u as a p-dimensional vector and uNi such that (uNi)j ̸= 0 for j ∈ N i

Assume A4. Denote u as a p-dimensional vector and uNi such that (uNi)j ̸= 0 for j ∈ N i. Then the dual norm of R(β) in the SGLIG model is upper bounded as follows: R∗(u) ≤ max i=1,...,p 1√ dmin ∥uG∥2. Proof. R∗(u) = max β {βT u} s.t. R(β) ≤ 1 = max V ( pX i=1 V (i)T uNi ) s.t. pX i=1 ατi∥V (i)∥2 + (1 − α) p di∥V (i)∥1 ≤ 1 ≤ max V ( pX i=1 V (i)T uNi ) s.t...

work page 2012

[11] [11]

(2019)) For chi-square random vari- ables X1, X2,

(Proposition S.5 Stephenson et al. (2019)) For chi-square random vari- ables X1, X2, . . . , Xp with d degrees of freedom and some constant c, P max i=1,...,p Xi ≤ c2d ≥ 1 − exp log(p) − (c − 1)2d 2 . Lemma

work page 2019

[12] [12]

Then the upper bound of R (∇L)2 is presented by setting c = r√n and r2 = log(p)+dmax dmax

Then by Lemma 7, P max i=1,...,p Vi ≤ c2d ≥ 1 − exp log(p) − (c − 1)2d 2 followed with R∗(∇L)2 ≤ σ∗σ2 n2dmin max i=1,...,p ∥Vi∥2 2 ≤ σ∗σ2 n2dmin c2dmax n with probability ≥ 1 − exp log(p) − (c − 1)2dmax 2 . Then the upper bound of R (∇L)2 is presented by setting c = r√n and r2 = log(p)+dmax dmax . Stephenson et al. (2019) provide explicit derivation of th...

work page 2019

[13] [13]

Assume (A1)-(A3), and let ΠMSG (·) represent the projection onto the subspace MSG

with a strictly positive regularization parameter satisfying λ ≥ 2R∗(∇L(β)), where R∗(·) is the dual norm of R(·) and ∇L(β) is the gradient of the loss function. Assume (A1)-(A3), and let ΠMSG (·) represent the projection onto the subspace MSG . Then, the error, ˆ∆ = ˆβ − β, will belong to the set: C(MSG , M ⊥ SG, β) := n ∆ ∈ Rp | R(ΠM⊥ SG ∆) ≤ 3R[ΠMSG (∆...

work page 2012

[14] [14]

Hence, we can summarize that : Qn(u) − Qn(0) = I1 + I2 + I3, where I1 d → 1 2 uT M u − uT W, while I3 + I4 d → 0 if supp(u) ⊆ J0 and I3 + I4 d → ∞ otherwise

Then we have |I4| → ∞ as n → ∞. Hence, we can summarize that : Qn(u) − Qn(0) = I1 + I2 + I3, where I1 d → 1 2 uT M u − uT W, while I3 + I4 d → 0 if supp(u) ⊆ J0 and I3 + I4 d → ∞ otherwise. Therefore, we can conclude that Qn(u) − Qn(0) d → D(u), where D(u) = 8 >>>< >>>: 1 2 uT M u − uT W, if supp (u) ⊆ J0, ∞, otherwise. Since ˆu = arg minu∈Rp Qn(u) = arg ...

work page 1996

[15] [15]

48 A.5 Comparisons of Finite Error Bound Between the SGLIG and DSRIG Scenario 1: Comparisons Under Assumptions of DSRIG The DSRIG uses the adaptive weights for the l2 penalty defined as τi = √di cov(X,Yi) , where |cov(X, Yi) ≤ 1| for ∀i ∈ {1, . . . , p}. Therefore, we have τ 2 i ≥ √di for ∀i ∈ {1, . . . , p}. Then it can be shown that τ 2 min ≥ dmin and 1...

work page 2019

[16] [16]

, p in assumption A4., which means τ 2 min ≥ dmin

The SGLIG model assumes τi is lower bounded by √di for i = 1 , . . . , p in assumption A4., which means τ 2 min ≥ dmin. However, The SGLIG model uses different adaptive weights for the l2 norm which is defined as τi = 1 cov(X,Yi) for ∀i ∈ { 1, . . . , p}, so there is no direct scale condition between τmin and √dmin when fitting the SGLIG 49 model to the s...

work page 2003

[17] [17]

(2019) used 135 volume measurements ( mm3) obtained from the Freesurfer segmented data as the predictors

and Stephenson et al. (2019) used 135 volume measurements ( mm3) obtained from the Freesurfer segmented data as the predictors. FreeSurfer is an image analysis suite used to process the MRI data collected in ADNI. It was developed by Hartig et al. (2014) and is available online (http://surfer.nmr.mgh.harvard.edu/). We are also interested in predicting the...

work page 2019