Fast Debiasing of the LASSO Estimator
Authors: Shuvayan Banerjee
James Saunderson
Radhendushka Srivastava
Ajit Rajwade
Venue: Accepted by TMLR ()

Abstract: In high-dimensional sparse regression, the \textsc{Lasso} estimator offers excellent theoretical guarantees but is well-known to produce biased estimates. To address this, \cite{Javanmard2014} introduced a method to ``debias'' the \textsc{Lasso} estimates for a random sub-Gaussian sensing matrix $\boldsymbol{A}$. Their approach relies on computing an ``approximate inverse'' $\boldsymbol{M}$ of the matrix $\boldsymbol{A}^\top \boldsymbol{A}/n$ by solving a convex optimization problem. This matrix $\boldsymbol{M}$ plays a critical role in mitigating bias and allowing for construction of confidence intervals using the debiased \textsc{Lasso} estimates. However the computation of $\boldsymbol{M}$ is expensive in practice as it requires iterative optimization.
In the presented work, we re-parameterize the optimization problem to compute a ``debiasing matrix'' $\boldsymbol{W} := \boldsymbol{AM}^{\top}$ directly, rather than the approximate inverse $\boldsymbol{M}$. This reformulation retains the theoretical guarantees of the debiased \textsc{Lasso} estimates, as they depend on the \emph{product} $\boldsymbol{AM}^{\top}$ rather than on $\boldsymbol{M}$ alone. Notably, we derive a simple and computationally efficient closed-form expression for $\boldsymbol{W}$, applicable to the sensing matrix $\boldsymbol{A}$ in the original debiasing framework, under a specific deterministic condition. 
This condition is satisfied with high probability for a wide class of randomly generated sensing matrices.
Also, the optimization problem based on $\boldsymbol{W}$ guarantees a unique optimal solution, unlike the original formulation based on $\boldsymbol{M}$. We verify our main result with numerical simulations.

Published in Transactions on Machine Learning Research (03/2026)
Fast Debiasing of the LASSO Estimator
Shuvayan Banerjec
bunerjecskhwaayan21 @gmail com com
Department of Mathematics
IITB-Monash Research Academy, Mumbai, India
James Saunderson
jomes.scwnderrom@ooona....
Deportment Electrical and Computer Systems Engineering Monash Unäversity, Melbourne, Australia
Radhendushka Srivastava
rsrivestora Dittbttnoriii..o.........
Department of Mathematics IIT Bombay, Mumbai, India
Ajit Rajwade
ajilor wct.sssc..bb........
Department of Computer Science and Engineering IIT Bombay, Mumbai, India
Reviewed on OpenReview: https: // wptp://eepprvvee...mmmee net/ forum? id= gEVPLLhoNI
Abstract
In high-dimensional sparse regression, the LASSO estimator offers excellent theoretical guar- antees but is well-known produce biased estimates. To address this, Javanmard & Mon- tanari (2014a) introduced method to "debias" the LASSO estimates for random sub- Gaussian sensing matrix A. Their approach relies on computing an "approximate inverse" M of the matrix A A/n n by solving convex optimization problem. This matrix M plays a critical role in mitigating bias and allowing for construction of confidence intervals using the debiased LASSO estimates. However the computation M expensive in practice as requires iterative optimization. the presented work, we re-parameterize the optimization problem compute "debiasing matrix" W AM directly, rather than the approx- imate inverse M. This reformulation retains the theoretical guarantees of the debiased LASSO estimates, they depend on the product AM rather than on M alone. Notably, we derive simple and computationally efficient closcd-form expression for W. applicable to the sensing matrix A in the original debiasing framework, under a specific deterministic condition. This condition is satisfied with high probability for a wide class of randomly gen- erated sensing matrices. Also, the optimization problem based on W guarantees a unique optimal solution, unlike the original formulation based on M. We verify our main result with numerical simulations.
1 Introduction
In high-dimensional sparse regression, where the number of predictors significantly exceeds the number of observations, the LASSO (Least Absolute Shrinkage and Selection Operator) is widely used method for variable sclection and estimation. By incorporating an & rogularization term, LASSO promotes sparsity in the estimated coefficients, enabling effective performance for sparse signal vectors even the number of predictors far exceeds the number of samples. The LASSO estimator has well-established theoretical guarantees for signal and support recovery (Hastic al. 2015). Despite
strengths, a well-recognized limitation of LASSO is its tendency produce biased estimates. This bias arises from the shrinkage imposed by the penalty. Consequently, the bias compromises estimation accuracy and impedes statistical inference tasks

---

Published in Transactions on Machine Learning Research (03/2026)
such construction of confidence intervals or hypothesis tests. These challenges are especially pronounced in high-dimensional regimes, where traditional inference tools fail due to high dimensionality.
To address these limitations, several methods have been developed to "debias" the LASSO estimator, allowing for valid statistical inference even in high-dimensional settings. Notably, Zhang & Zhang (2014) introduced a decorrelated score-based approach, leveraging the Marush-Kuhn-Tuckee (KKT) conditions of the LASSO optimization problem to construct bias-corrected estimators. Their framework relies on precise estimation of the precision matrix (inverse covariance matrix), which can be computationally challenging and sensitive to regularization choices. Similarly, Van de Geer et al. (2014) proposed methodology rooted in node-wisg wise regression, where each variable is regressed on the remaining variables to estimate the precision matrix. While effective, this method is computationally intensive. This may limit its applicability, particularly in scenarios where the design matrix lacks favorable properties like sparsity of the rows of the precision matrix.
Javanmard Montanari (2014a) introducod simple yet powerful approach that constructs dcbiased LASSO estimates using an "approximate inverse" of the sample covariance matrix. Their method avoids direct precision matrix estimation and instead employs an optimization framework to compute debiasing matrix M that corrects for bias while ensuring asymptotic normality of the debiased estimates. A key advantage of this method is its applicability for random uub-Gaussian sensing matrices, enabling valid inference across a broad range of high-dimensional applications.
In this work, we build upon the technique of Javanmard & Montanari (201aa), addressing one of its primary computational bottlenecks: the optimization step required to compute the approximate inverse M. By refor- mulating the problem to work directly with the "weight matrix" W == AMM,, we entirely eliminate the need to solve this optimization problem in many practical cases. Our proposed reformulation leverages the insight that the theoretical guarantees of the debiased LASSO estimator depend on the product AM rather than the individual debiasing matrix M. By shifting the focus to the "weight matrix" W := AM we simplify the optimization problem while retaining all theoretical propertics of the original framework. Under certain deterministic assumptions, we provide a simple, exact, closed form optimal solution for the optimization problem to obtain W. We show that this assumption is satisfied with high probability for different. popular ensembles of sub-Gaussian sensing matrices, under the additional condition that the elements of the rows of A are weakly correlated. In practice, sensing matrices with uncorrelated entries are commonly used in many applications (Duarle et al., in sparse regression (Hastie et al. optimization step required to compute M, significantly improving runtime efficiency. It is applicable in many natural situations, including sensing matrices with i..d. isotropic sub-Gaussian rows (such as i.i.d. Gaussian, or i.i.d. Rademacher entries).
2008 Liu et al. 2013) and are also widely used in many theoretical results 2015 This closed form solution eliminates the computationally intensive
Throughout this paper, we denote matrices by bold-faced uppercase symbols, c.g., A. If A is an n x p matrix then a, E RP denotes the 7th row of A, thought of as a column vector. Similarly if A is an n x p matrix then a.j E IR" denotes the 5th column of A, again thought of as a column vector. Vectors are denoted by bold-faced lower case symbols, e.g., w. The ith entry of a vector is denoted w; E R. The identity matrix of size p x p for any positive integer p is denoted by Ip. and its tth column vector is denoted by eg. For a positive integer p, we use the shorthand [p] = {1, (1,2....,P)) ,P). For a vector w E R we denote the norm by ||l||- == EE [w, if f q < oo and the norm by ||w||s == maXie[m] |Ws].
Notation:
2
An Overview of the Debiased LASSO
We consider the high-dimensional linear model
y A,9* 77
where 3² E Rp s-sparse signal (i.c., s = 11 l1o where s < p), A is a n anxpp design/sensing matrix (where n < p), and y E R" is the measurement vector. Also, n E R is an additive noise vector that consists of independent and identically distributed elements drawn from N (0, d²), where o² is the noise variance.
2

---

Published in Transactions on Machine Learning Research (03/2026)
The LASSO estimate Ax of the sparse signal 8 is defined the solution to the following optimization problem:
1 Dx arg - yy AB13 A|1011,
(2)
2n
where > O is a regularization parameter chosen appropriately. The LASSO estimator is known to be a consistent estimator of the sparse signal 8 under the condition that the sensing matrix satisfies the Restricted Eigenvalue Condition (REC) (Hastic al. 2015 Chapter 11).
The LASSO estimator is well-known to produce biased estimates, i.e, E(Bx) * 3 where the expectation is computed over noise instances. This bias arises from the & regularization term, which induces shrinkage the estimate Bx. Moreover, there is no known method to compute confidence interval of 3 directly from ax
To reduce this bias and also construct confidence intervals of Ø", Javanmard & Montanari 2014a introduced debiased LASSO estimator Ba, defined follows:
:= Ax MA (y ABx).
n
Here M is an approximate inverse of the rank deficient matrix Σ == A A/n, computed by solving the convex optimization problem given in Algorithm ! The parameter l in A controls the bias of the
debiased LASSO estimator given in 3 Ideally one should choose the smallest μ for which (4) is feasible.
Algorithm Construction of M (from Javanmard Montanari (2014a))
Require: Design matrix A, μ (0,
Ensure: Debiasing matrix M
1: Compute: Σ = A/n.
2: For each j E [p], solve the following optimization problem to compute column vector m.j E RRP:
minimize m m Sm
subject to [Em - ee S p,
(4)
where ej is the jth column of the identity matrix I,, and μ E (0,
3: Assemble M as M = ..m.--/m,p)
4: the optimization problem infeasible for any j, set M == Ip.
The theoretical properties of Ba are applicable to sensing matrix A with the following properties:
D1: The rows a1 ag....., an, matrix A are independent and identically distributed zero-mean sub- Gaussian random vectors with covariance Σ a.a Furthermore, the sub-Gaussian norm K = Σ 1/2,, 1l is a finite positive constant.
D2: There exist positive constants 0 < Cmin I Cmax: such that the minimum and maximum eigenvalues mmin E), 0max (E) of Σ satisfy 0 < Cmin S Gmin (2) S Tmax (E) S Cmax
00.
Theorem 7(b) of Javanmard & Montanari 2014a shows that the optimization problem in is feasible with high probability, for sensing matrices satisfying properties D1 and D2, as long μ 4
Cmaz logg
Cain
If u is O loge and is w((s log ))) then Theorem 8 in the aforementioned paper shows that the bias of the debiased LASSO estimator goes to and Vj E [p], vnlBa B3) is asymptotically zero-mean Gaussian with variance o² omm J.m..-
'Te sub- Gaussian norm of random variable n, denoted by || is defined as || = supg21 q 1/2 (E), For random vector R", sub Gaussian norm is defined as ||llleg supyes 1 |1y a2llgg where gn-1 denotes the unit. sphere R
3

---

Published in Transactions on Machine Learning Research (03/2026)
3
Re- parameterization of the Debiased LASSO
The debiased LASSO estimator in can be rewritten in terms of the weight matrix W == AM as:
3 - W y ABx).
n
The re-parameterizztionn does not affect the debiasing procedure described earlier. Thus, any theoretical guarantees established using M extend to those using W.
We now produce a reformulated problem in 16) using W, and show that is equivalent to the original optimization problem in Algorithm [ Using the relationship W AMT we can rewrite m as w.y == Am.j. Making this substitution, the objective in (4) becomes m [mm, = " , and the constraint
Em,- e $ μ (where ej is the jth column of the identity matrix) becomes [ w. - -e, e S p.
This change of variables suggests the following reformulated optimization problem 16 for the j column of W:
9
minimize
1 w w. J n S..
subject
1. A ; eg
(6)
In fact, the jth reformulated problem 16 and the jth original problem 4 are equivalent in the following sense: If m.j is feasible for 4 then w.j = Am. is feasible for (6) and w _ = m.JEm.j. Conversely, suppose that w.j feasible for 6 If At is psoudo-inverse A, then m.j == A A is feasible for 44) since Em, A Am. 1 wn. Moreover, n , = m m m_mmm,,, so both have the same objective values, establishing that (4) and 16 are equivalent. This reformulation provides an equivalent separable problem for each column W, maintaining all theoretical guarantees while simplifying the representation
of the debiasing procedure.
The reformulated problem (6) has unique optimal solution because the objective function is strongly convex with convex constraints. In contrast, the original problem 4 does not have a unique solution. Indeed if m.j
is any solution to then we can add to it any element of the nullspace of A to obtain another solution
to
3.1 A Closed-Form Solution for the Debiasing Matrix W
In this section, we demonstrate that, for suitable choice of μ., the optimal solution to the problem (6 can be computed in closed form for a sensing matrix whose minimum column norm is strietly positive (which is true with probability for random matrices). To derive this result we write down the Fenchel dual of (6) and appeal to weak duality. In particular, we explicitly find primal and dual feasible points with the same objective value, certifying that both are, in fact, optimal.
Thcorem Let A be a n. X p matriz with no column equal zero. Define p(A) maxi laTa,l ||a,
The
optimal solution of (6 is given by
w. = n11 p))
a for all j E [p]
(7)
aa 12
if and only if T 11
The proof of this theorem given Appendix A.1 For notational simplicity, we will denote p(A) by p the rest of the paper. We will provide a brief overview of the proof here.
Overview of the proof of Theorem 1 The proof shows that the closed-form expression in 7 is optimal
for the convex program ( when P < 1. Under this condition, the candidate vector satisfies the t feasibility constraint given in 66 because the coherence bound ensures (1 =)p S J.. Its objective value
4

---

Published in Transactions on Machine Learning Research (03/2026)
be computed in closed form and is given by (1-ي)² ||,,1//n' Using Fenchel dual reformulation, the dual problem is explicitly provided, and suitably chosen dual vector attains a dual value that matches the primal value achieved by (7) By weak duality, this equality certifies optimality. Conversely, if p. < p/ (1++/) +p,, the solution in (7) violates the feasibility constraint, and when μ > 1, the zero vector is the unique minimizer which is trivially optimal. Therefore, the stated range of p is necessary and sufficient for the optimality of the exact solution.
Remarks:
This theorem eliminates the requirement. to execute an iterative optimization algorithm to obtain W (or an iterative optimization algorithm to obtain M). This is because given A, one can directly implement the optimal solution of Alg. [ in the form (7) for all j E [p]. This speeds up the implementation of the debiasing of LASSO for the ensemble of sensing matrices that satisfy the conditions of Theorem 1 Likewise, our approach will also be significantly faster than the debiasing approach presented in Van de Geer et al. (2014), which explicitly estimates the precision matrix (inverse covariance matrix) of the rows of A via a series of different LASSO problems, each solved iteratively see equations 7,8,9 of Van de Geer et al. (2014).
The condition p1(+pp) S م 1 necessary and sufficient for the closed-form expression in 77 be optimal (). However, it is possible that (6) is feasible for values of μ that smaller than p/ p (++)). p).
In such situation, the optimal solution to 16 is not given by 7 This is empirically illustrated in Sec. 5.1 Tn the context of the debiased LASSO, Theorem 7(b) of Javanmard & Montanari (2014b)
shows that the choice of μ == Op(v loog//gg//// makes the optimization problem in (4) feasible for a sensing matrix A that satisfies conditions D1 and D2. For a wide class of sensing matrices, we show Theorem that p1 p) Or /lgg//n.
3. The quantity Ihe can be computed exactly by using the definition of given in Theorem given sensing matrix A. Furthermore, the distribution of Tép can also be estimated via simulation for given any n, p, and E corresponding the sensing matrix A.
The solution in 7 is the optimal solution even when we choose = 1. The optimal solution this case is the trivial solution wy = O. However practice, one always chooses μ to be small, and hence this specific situation does not arisc
3.2 Concentration bounds of e
(v log
then Vj E [p), vn(la (3)) is asymptotically zero-mean Gaussian when the elements of are drawn from .N (N a²). For specific classes
As mentioned carlier at the end of Sec. 2 if is O
and is
of random sensing matrices, show in Theorem 2 that The coy log with high probability for some
constant This implies that for these random sensing matrices, the choice :== (v
log
ensures both
the following: (i) asymptotic negligible bias of the estimator Pa given by (5) when is w((s log p) and (#) fulfillment of the sufficient condition 1he S μ for the debiasing matrix W to be computed in closed-form.
If A satisfies an additional mild assumption given in Theorem then Ife S &cov logg holds with high probability.
Theorem Let A be a n x sub- Gaussian rows and sub- Gaussian norm K. :=
p dimensional matria with independent and identically distributed zero-mean
1 11/2,,,llls - where n < and Σ = Ela,,a,, T]. Let [Ety]
p be as defined in Theorem Let 7 2
Cmin 12 logp
maxi Ess If A obeys properties D1, D2 and
x Cmax
5

---

1+p Published in Transactions on Machine Learning Research (03/2026)
80 11
maya log for some e E (0, 1), then
C
min (1-c
p 1+p s2v2+ x² Omax
logp
3
2 1
c Cmin
2p
Furthermore, c
22/2+2, 1) and μa = (2V2 + 7) x² Cuax 4v/2ty"
logp then with high probability, is feasible
Cuin
and the optimal debiasing matrix W in 6 is given by
The proof of Theorem 2 is given in Appendix A.2
Remarks:
E is a diagonal matrix (i.e., entries of the sensing matrix A are uncorrelated), then we can choose
= 0 and the upper bound of p/(1 + p) in S reduces to 2/2*² Cmax
logg for any constant
Cain V
c 1). If not diagonal matrix, the parameter y represents a degree of dependence between the elements of the rows of the matrix A.
2. The condition c E (2V2+ + 7 H² Crax Cmin
22/+z 1v247'
1) in Theorem ensures that when n >
22 (1-c)² log p and f ==
SC2 SC
logg
then we have j < n
3. In practice, one tends to choose small value of js for debiasing the LASSO estimator. Given n, p and A Σ, the exact distribution Itp can also be estimated with high precision through simulations. One may also choose j to be slightly larger than the maximum support of the distribution of __. This produces a simple and elegant way to choose μ in practice. Given a fixed A, is easy to compute p. by definition of p in Thcorem 1 So we can simply set μ ı= 1+p
P since smaller (but feasible) values
of μ are desirable in (6) In Sec.5.4 we observe that this empirical choice of l is smaller than the
choice of (2v2 + 7)
Cmax
log.
Caun
4 Relation to Recent Developments in LASSO Debiasing
The LASSO debiasing literature has seen many recent developments in the statistics as well the AT/ML communities. For instance, the theory of the debiased LASSO has been extended to handle generalized linear models in Vazquez & Nan (2025 ; Xia et al. (2023). Applications of the debiased LASSO (extended to handle the total variation image prior) to compressive reconstruction of magnetic resonance images have been recently explored Hoppe et al. (2024b). Bootstrap approaches to further diminish the bias value in low-sample regimes have been explored in Li (2020). Along similar lines, a learning based technique for further reduction of the bias in small-sample regimes has been recently explored in Hoppe et al.) (2024a), where it is also shown how to incorporate debiasing for data-driven approaches such unrolled neural networks. Debiasing techniques have also been extended to handle sparse quantile regression in Yan et al. (2023). Note that all these techniques in principle require computation of either the approximate covariance matrix M or the debiasing matrix W, after which various other steps in their approach are carried out. Since our work this paper proposes technique to speedily compute W = AM is clear that it can be readily incorporated to speed up the key stop of estimation of M or W in cach of these aforementioned approaches.
5 Empirical Results
The MATLAB codes for the simulations in this section and also those from Sec. G are available at https: //gi thub. hctp://uwaaaaaa///aaaaaaaaaaaaaa

---

Published in Transactions on Machine Learning Research (03/2026)
5.1 Difference between the exact closed form solution We and the solution of the optimization problem in 6 given by W. for varying choices of
1+p S < then the exact closed form solution of represented by We the same the solution of the optimization problem given in ( represented by W.. o- In this subsection, we investigate the difference between W. and We for μ < TTp well in the range _ S p 1.
Aim: In Theorem we show that if
We report the diffcrence between We and W. in terms of the Relative Error given by (IW ([W--W-llr) for μ 0.2, 0.21, 0. 0.600.
Sensing matrix properties: For this experiment, we fixed 80, p 100. We this experiment for two different p sensing matrices A with elements drawn from: (1) i.dd Gaussian and, (2) i.i.d. Rademacher. In Figure we plot la vs ([Wo-Wele) W.le) for both of these matrices on log scale. The exact [Well value of A is given by a black vertical line in each case,
Itp
Observation: We see that for both the plots in Figure 1 the relative error decreases with increase in j for μ Ttp For 2 T-p. the relative error is very small with fluctuations primarily due to the solver tolerances in Isqlin when computing Wo. Furthermore, the decrease in relative crror is sharp after the P value μ crosses 14p"
10a
10"
1
10
relative arror
relative error
10
10-
10
10
10
10.
10-10
10-10
0.2
0.35
0.5
0.55
0.35
0.5 0.55
Figure 1: Line plot of y relative error wll (in log scale) for two 80 100 dimensional sensing matrices: (left) iidd. Gaussian and (right) ii.d. Rademacher. The exact value of _ is given by the black vertical line. The value of Itp 0.327 for the Gaussian sensing matrix (left) and 0.298 for the Rademacher sensing matrix (right). Here, W. the solution of the optimization problem in 6 and W. is computed as in
WellF
5.2 Comparison of debiasing performance using the exact solution W. and the choice M dE-1
In this subsection, we compare the sensitivity, specificity of the debiased estimate Pw. obtained from using the exact closed-form solution We, and the debiased estimate Bax given in Equation (17) of Javanmard & Montanari (2014b) with the debiasing matrix M == dI (note that M is an approximato inverse of the empirical covariance matrix Σ see Sec. 2 where d := (1 ||B llo/p) We further compare the ratios of the empirical total variance (ETV) and asymptotic total variance (ATV) of ßw. and B 1. The ATVs of the jth element of 3w. and B are respectively given by 1, E w.,. and of - "ig- The empirical total variance (ETV) of the debiased LASSO estimators is obtained using 100 simulation runs over dillerent. instances of n with varying (250, 350,00);. = 500, = 5 where the signal 3 was generated in the same manner as described in the beginming of this section. In those simulations, the rows of A are generated as p-dimensional i.id. random vectors from (, Σ) for three different choices of Σ given as follows:
0.01,
1. Diagonal Matrix: Σ, I²I, with choice o² 1.

---

Published in Transactions on Machine Learning Research (03/2026)
2. Banded Equicorrelated Matrix: En with (i, entry follows.
0². if [p]:
Σ2,5 = S |f i E [p] with choices (= 0.1, b= and 0² = 1.
0,
otherwise
3. Equicorrelated Matrix: Σ3 = o2 - r) + (1ppp] with choices ( 0.1 and o² = Here, 1p denotes the p-dimensional vector of all ones.
The diagonal covariance matrices are widely used in compressed sensing Candès et al. (0066) The chosen banded equicorrelated covariance matrix (Σ2) is special case of symmetric circulant matrix, which has been explored by Javanmard & Montanari (2014a) in the context of debiasing the LASSO estimator. Further, the equicorrclated matrix Σg has motivation in compressed sensing well, to express tross-talk-give by the lerm (11 belween different elements of a sensor array. In single-pixel cameras (a common archilecture in compressed sensing) Duarte et al. (2008), the term the term Ç1,1p models global illumination changes (similar to background interference) which bring in weak correlation, so that the jth row of the sensing matrix can be effectively expressed by a' /1- + V
In Tables and 3 we present results for each of these covariance designs comparing Sensitivity and Specificity, for both debiased estimates (using W. and 1) We also present the ratios of the ATV and ETV for these estimates.
Sens(Bw.)
Spec(Bw.))
Sens(Bux->))
Spec(Bax
ATV(Sw.)
ETV(S
n
ATV(3 (x-)
ETV (x1x))
250
0.7145
0.8972
0.7209
0.8653
0.2819
0.3863
350
0.8554
0.9719
0.8126
0.9233
0.3882
0.5182
500
0.9985
0.9992
0.9486
0.9492
0.4699
0.6075
Comparison sensitivity, specificity, and ATV, ETV ratios for the debiased estimates åw. and faz= across different sample sizes n E [200 50 500) for an uncorrelated Gaussian design matrix. The fixed parameters are 500, fa = 0.01, 8 = 5, r=
Table Diagonal Matrix E,: (see Sec. 5.4
Sens( (e.)( B..)
Spoc(âw.)
Scns( (
Spec(8a2>
ATV(Bw.)
ETV
ATV( (x-11)
ETV(
250
0.7392
0.8871
0.6975
0.8387
0.2573
0.3982
350
0.8833
0.9562
0.8136
0.8865
0.3142
0,5961
500
0.9715
0.9854
0.9006
0.9216
0.3924
0.7269
Table 2: Banded Equicorrclated Matrix Ezi (sce Sec. 5.4 Comparison of scnsitivity, specificity, and ATV, FTV ratios for the debiased estimates Pw. and Pax-1 across different sample sizes n = [200 : 50 500] for correlated Gaussian design given as a bandwidth-3 matrix with Dij o² 001,, -i 3, and zero otherwise. The fixed parameters p = 500, fo = 0.01, = 5, 5
Sens(pw.]
Spec(3w (
Sens(
Spec (Be
ATV ATV(O -x-1) 0.2724
ETV (.w.)
n
PTV(BEE-
250
0.7275
0.8655
0.6855
0.8152
0.3892
350
0.8112
0.9216
0.7908
0.8872
0.3433
0.4844
500
0.9466
0.9573
0.9212
0.9319
0.4147
0.5795
Table 3: Equicorrelated Matrix Eg: (soc Sec. 5.4 Comparison of scnsitivity, specificity, and ATV,
ETV ratios for the debiased estimates 3w. and 3 3ax-1 across different sample sizes n [200 50 500] for correlated Gaussian design given as a bandwidth-3 matrix with Eij = o² 0.1, li - jl S 3, and zero
otherwise. The fixed parameters are = 500, fo = 0.01, s 5,

---

Published in Transactions on Machine Learning Research (03/2026)
2 and 3. it is evident that the debiased estimator Sw. consistently outperforms faz= in terms of sensitivity and specificity, with the advantage being more pronounced for smaller sample sizes and gradually diminishing as n increases. This performance beneîit is not surprising because in our approach, the matrix W is specifically designed to produce a debiased estimator of minimum variance, unlike the choice of M == dE- which only provides debiasing. Furthermore, the debiasing properties of M := : have only been established for Gaussian uncorrelated designs in Javanmard & Montanari (2014b), whereas our approach is applicable to a much wider range of matrices. Moreover, our approach does not require knowledge of Σ, which may not be available and is hard to estimate even for uncorrelated designs because n < p. Lastly, our approach does not rely on the lo norm of the LASSO estimate.
From Tables
The introduction of correlation in the design matrix leads to an overall reduction in both sensitivity and specificity at lower n, but this gap narrows down with larger n. Furthermore, the variance ratios remain below unity across all settings, indicating that Sw. achieves lower empirical and asymptotic variances, with the ratios increasing steadily in n, reflecting greater stability. Overall, ßw. demonstrates superior efficiency and robustness to correlation compared to BIg-..
5.3 Validity of the exact solution
The debiased LASSO can be used determine the support of the unknown vector 8 by using statistical hypothesis tests derived using LASSO debiasing theory. We aim to estimate the support using p hypothesis tests (one per element of (8)) based on the debiased LASSO estimates using the weights matrix W obtained from the optimization problem in (6) (denoted by Wo), and that obtained from the closed-form expression 7 (denoted by We), for varying number of measurements n. The aim is to also compare these support set estimates with the ground truth support set, and report scnsitivity and specificity values (defined below). We will further show the difference in the run-time for both methods.
Aim:
Signal Generation: For our simulations, we chose our design matrix A to have elements drawn indepen- dently from the standard Gaussian distribution. We synthetically generated signals (i.e., 8²)) with p = 500 clements in each. The non-zero values of 3 were drawn i.i.d. from U(50, 1000) and placed at randomly chosen indices. We set 8 := |1,8% Ilo = 10 and the noise standard deviation a = 0.05 Ei |at,3*|/n. We varied {200, 250, 300, 350, 400, 450, 5003. We chose μ p/ pp+ + 1) where p was computed exactly given the sensing matrix A.
Sensitivity and Specificity Computation: Let us denote the debiased LASSO estimates obtained using a matrix W by Pa,w. We know that asymptotically 3a.w (j) M (S3.. oZw ww ,//²) for all j E [p]. Using this result, Sa, was binarized to create vector bw in the following way: For all j E [p], we set (j) = if the value of (5) was such that the the hypothesis Ho.j /3 O was rejected against the alternate Haj 83 # 0 at 5% level of significance. bw()) was set to 0 otherwise. Note that for the purpose of our simulation, we cither have W = W. or W = We. The binary vectors corresponding to these choices of W are respectively denoted by bw. and bw.-
(j)
A ground truth binary vector b* was created such that b3 := 1 at all locations j where 83 # 0 and 6ū := 0 otherwise. Scnsitivity and specificity values were computed by comparing corresponding entrics of b* to those in 6w. and bw Considering the matrix W, we declared an element to be a true defective if bj = and (j) = 1, and false defective if b3 = O but (j) + O. We declare it to be a false non-defective if by = 0 but (J) F 0, and a true non-defective if 83 = 0 and (w) = 0. The sensitivity for 3 is defined as (# true defectives)/(# true defectives + # false non-defectives) and specificity for 9 is defined as (# true non-defectives)/( true non-defectives + # false defectives).
Results: For obtaining W., the optimization routine was executed using the Isqlin package in MATLAB. The sensitivity and specificity were averaged over 25 runs with independent noise instances.
In Table 4 we can see that the sensitivity as well as the specificity of the hypothesis tests for W. and W. are equal. We further report. the relative difference between W. and Wa in the Frobenius norm. We
9

---

Published in Transactions on Machine Learning Research (03/2026)
sensitivity
specificity
time (in s)
W
W
Wo
W.
W.
Wa
W.-W.il W.lle
W
200
0.6742
0.6742
0.8592
0.8592
3.88 x 10a
1.11 X 10
6.68 x 10 10-10
250
0.7229
0.7229
0.9063
0.9063
5.22 x 102
1.72 x 10
2.31 x 10
300
0.8071
0.8071
0.9427
0.9427
3.29 x 102
2.25 10-3
2.73 x 10
350
0.8554
0.8554
0.9719
0.9719
4.77 10a
3.88 10-3
2.56 10
400
0.9275
0.9275
0.9855
0.9855
5.59 10a
7.82 10-3
4.76 x 10
450
0.9781
0.9781
0.9909
0.9909
7.15 x 10a
4.27 x 10
5.29 X 10
500
0.9985
0.9985
0.9992
0.9992
8.03 x 102
7.56 x 10-2
8.22 X 10-7
Table 4: Sensitivity and Specificity of hypothesis test using debiased estimates obtain from W. (optimization method) and We (closed-form expression from 7 with its corresponding runtime in seconds for varying number of measurements. The fixed parameters are = 500, s 10, = 0.05 E.= 1/n. We set
== pp(p + 1) where p is computed exactly for the chosen sensing matrix A.
sensitivity
specificity
time (in s)
Wo
We
W.
We
W.
We
WW- T...e.lle
[Wale
0.2
0.9586
0.9544
0.9942
0.9901
8.44 X 10
7.76 10 -3
2.24 10
0.25
0.9531
0.9502
0.9872
0.9855
6.91 x 102
8.72
10-3
7.62 x 10-2
0.3
0.9475
0.9475
0.9921
0.9921
5.59
10a
8.12
10-3
3.39 10
0.35
0.9354
0.9354
0.9891
0.9891
5.42
102
7.83
10-3
6.312 10-7
0.4
0.9275
0.9275
0.9855
0.9855
5.77
102
7.56
10-3
2.08 10-8
0.45
0.9102
0.9102
0.9792
0.9792
5.98
102
7.49
10
4.55 10-7
Table 5: Sensitivity and Specificity of hypothesis tests using debiased estimates obtained using W. (opti- mization method) and We (closed-form expression from (7)) for varying choice of μ.. The corresponding run-times for estimating W. and W. from Rademacher sensing matrix A, are also shown. The fixed parameters are = 500, n = 400, = 10, o = 0.05 Σ (d,,8"/no The exact value +4, 0.298 where p is computed exactly for the chosen sensing matrix A.
10

---

Published in Transactions on Machine Learning Research (03/2026)
can clearly see that the difference is negligible, which is consistent with Theorem Furthermore, we see that using the closed-form expression in saves significantly on time (by factor of at least 10*). While the computational efficiency of the iterative approach can be improved by developing specialized solver for problems of the form (6) no itcrative method is expected to outperform directly computing the simple closed-form expression
we observe that both debiasing matrices Wo and We exhibit almost identical sensitivity and specificity for the hypothesis tests across wide range of μ values greater than equal to 0.2 to 1. This range was chosen because we observed that for the choices of μ < 0.2, the optimization problem 4 was often not feasible. For μ > p/(1 p) = 0.298, the scnsitivity and specificity of the debiasing methods with W. and Wa was the same (up to numerical tolerances in the optimizer) which is consistent with our theory. For μ > 0.45, the sensitivity and specificity of LASSO debiasing with both W. and We was below 0.9 (not shown in the table), but it remained identical for both methods. For 0.2 < μ < 0.298, the sensitivity and specificity values with the two mothods were similar even though not identical. In all cases, however, the major distinction between the two methods was computational time, as computing W. look more than 550 seconds whereas We was obtained in a few milliseconds. Given this dramatic speed-up and the similar statistical performance, the closed-form We offers highly practical and efficient alternative to the optimization- based solution W..
In Table 5
5.4 Empirical Distribution of p/(1 + p)
In this subsection, we will show that the support of the distribution of e is smaller than the choice of
Itp
;= = (2 + 7) m" Coax
Cmin V logg given by Theorem for the different chosen covariance matrices E1 E2. Σg
defined in Sec. 5.2 We chose p = 500 and n |250, 350, 5003. For each configuration, we generated 1000
independent n x p matrices A, with rows sampled i..dd from (0,
[a [T]a The normalized histograms of p/(1 + p) For each realization of A, we computed p(A) = maxifj T|a 5ll;"
based on 1000 simulation runs are shown in Figure 2 The top, middle and bottom rows of Figure 2 re- spectively correspond to the covariance matrix E,. Σg and Σg, whereas the left, center and right column respectively correspond to n = 250, 350 and 500. Each plot. is overlaid with a red vertical line showing the d Cuaax
bound μ = (2v2+>)
logg given in Theorem 2 Tdeally, we would like to choose y as small as pos-
Crat n
sible and e to be as large as possible. Therefore in our experiments, we chose 7
Cana x=Cous
[Eg] n maxişj Ez logp
Furthermore, under the assumption n >
802 8C
logp, we chose = - 2v x²
Cmax
logg
Cmin
n
In Figure 2 we observe that the span of the normalized histograms shrinks and moves towards the origin for all the chosen covariance matrices. This indicates that - lends to zero as the sample size increases.
Figure 2 also shows that the upper bound on Ife obtained from Theorem 2 is conservative (in terms of constant factors) for smaller sample sizes. We also observe that the probability density of I+p depends on the dependent structure of Σ. Therefore, given the values of n, p and E, one may choose μ which is slightly larger than the maximum value of the support of the distribution of _ in practice, which can be obtained using simulation before performing the debiasing.
T+p
6 Experiments on Compressive Image/Video Reconstruction
We further validated the use of our fast debiasing approach for image reconstruction from noisy compressive measurements given three different compressed sensing architectures (that is, realistic models for A): the Rice Single Pixel Camera for compressive imaging Duarte et al. (2008), the coded exposure snapshot camera for compressive video acquisition Liu et al. (2013) and the coded aperture snapshot spectral imager (CASSI) for hyperspectral image acquisition Kittle et al. (2010). There already exist a plethora of compressive image roconstruction techniques, both classical Foucart & Rauhut (2013) and deep learning based Kulkarni et al.. (2016). Here, our aim is to provide proof of concept that fast debiasing is applicable to realistic sensing matrices; our aim here is not to beat. the state of the art However, compared to the existing techniques, our
11

---

Published in Transactions on Machine Learning Research (03/2026)
250
350
500
0.4
0.4
0.4
density
0.3
E1
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.4
0.4
density
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0.2
0.6
0.8
0.2
0.4
0.2
0.4
0.4
0.4
0.4
density
0.3
0.3
0.3
E3
0.2
0.2
0.2
0
0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
pl(1+p)
pl(1+p)
pl(1+p)
Figure Histograms of p/(1 + p) based on 1000 simulation runs for p = 500 and n E {250, 350, 500). Rows correspond to different covariance structures (diagonal, banded equicorrelated and equicorrelated), while columns correspond to sample size n. The red lines indicate the theoretical benchmark of μ := (2v2 + 7) Cmax logg by choosing c = 2v222 designs.
Caux Cuin
Cmin logg and 7 C2Cama
[Eul for all log maxtzj Egg
Cmin a
presented approach is unique in its ability to provide quantification of the uncertainty in the reconstructed pixel values, an aspect which the aforementioned techniques do not cover.
Consider a noisy measurement vector y E R the form y = $f + TTO + n, where n is an additive noise vector whose elements are drawn independently from .V o2, E Rexp sensing matrix i.idd. sub-Gaussian distributed entries, and E Rexp an orthonormal basis in which the image f is sparse or 'weakly sparse' that f = f0, where E IP vector of coefficients of which a small number have large magnitude and the majority are cither zero (sparse O) or close to zero (weakly sparse 0). The aim is to reconstruct 0, and thus f. from Φ V in the compressive regime where & p. For this estimation task, the LASSO is used: = arg ming |/ TTO0 2 + A|[O||1 This is followed by debiasing step of the form 0a 0 +++1V W
v where the debiasing matrix W (1 μ with being exactly defined in Thcorem with A = TΦ. We particularly note that in this case, we first estimate O since the image f E IRP is not sparse in the canonical basis but is (weakly) sparse in the basis w. A typical choice for w is the 2D discrete cosine transform (DCT) basis, though many other choices such as wavelets, shearlets, etc., are also possible. We note that Theorems [ and [ are applicable to the matrix
Φ for the following reasons: (1) As no column of & equals zero, neither does any column of $ and hence Theorem applics. (Wi + has zero-mean aub-Gaussian entries and V has bounded entries. Hence the entries of t v are also
12

---

Published in Transactions on Machine Learning Research (03/2026)
RRMSE
SSIM
Cov. Prob. (Edges)
Cov. Prob. (Non-Edges)
Barbara
0.0325
0.932
0.72
0.93
Cameraman
0.0281
0.948
0.68
0.91
Moon 0.0204
0.971
0.79
0.95
Male
0.0357
0.918
0.66
0.92
Table 6: Reconstruction quality metries for grayscale images of size 256 x 256 using Debiased LASSO. The metrics reported include RRMSE, SSIM, and average coverage probabilities computed separately on edge and non-edge pixel locations based on the Canny edge detection method.
zero-mean sub-Gaussian and they have the same covariance matrix as the rows of 4. Hence if the entries of d are i.i.d. Gaussian, then the rows of u remain independent. Hence Theorem 2 applies. Even though the independence assumption may not strictly hold for the rows of 4 for & from other distributions (e.g. Rademacher or Bernoulli), we have observed excellent empirical results with debiasing even for such cases.
6.1 Image reconstruction for the Rice Single Pixel Camera Model
We simulated noisy compressive measurements from four commonly used images of size 256 x 256 via a & matrix with i..dd entries drawn from a Bernoulli distribution with success probability 0.5, which is in tune with the architecture of the celebrated Rice Single Pixel Camera Duarte et al. (008). In our exper- iments, we used n = 20, 000 measurements for p = 256". The original images, and their reconstructed versions via the LASSO and the debiased LASSO, are shown in Fig. B. After debiasing. the variance of ~ T Whi Since f = NO, the variance of the ith pixel of the ith estimated coefficient is given by @2
the estimated image (i.e. AA) is given by W' I... These variance values form a quantitative measure of the uncertainty inherent in the reconstruction. To illustrate this further the reconstruction experiment for each image was repeated K 50 times using different realizations of the noise n, The total number of times N; (out of K)) for which fi (the value of the ith pixel of f) resided in the interval
/1-21-0/201 v WW Whr Jús f
V W Ti was recorded. This is the confidence inter-
val of probability 1 a for fú, and 21-a/2 is the percentile of a standard normal distribution at level 1 1- a/2. The values of N,/ termed coverage probabilities across all i E 1I.2....,?! were recorded and plotted as an image in Fig. 3 As is clearly seen in the fourth column of Fig. 3 these probabilities are very high for pixels in smooth regions and they are the least for pixels lying on edges. This is because the 2D DCT is more efficient in representing smooth regions as compared to discontinuities. The ratios along edges can be improved by using senssiive bases such shearlets Kutyniok & Labate (2012) or learned overcom- plete representations Aharon et al. (2006), but we leave full investigation of these aspects to future work as they are not central to the main theme of this work.
110--ā11a where Ô denotes the estimate We report the Relative Root Mean Square Error (RRMSE), defined as ||0-|| of the true parameter 0" Along with RRMSE, we present the Structural Similarity Index Measure (SSIM), the average coverage probability over edge pixels, and the average coverage probability over non-edge pixels. These quantitative metrics, summarized in Table G correspond to the Debiased LASSO estimates for all four benchmark images. The small RRMSE values and high SSTM scores collectively indicate that the the reconstruction quality very good. Furthermore, the coverage probabilitics display the expected spatial behavior: coverage is considerably higher in smooth, non-edge regions and lower in edge regions, consistent with the qualitative observations in Fig. 3
Image reconstruction with + having equicorrelated entries: Another set of results was obtained where the i.i.d. Bernoulli model for & was replaced by the equicorrelated model from Sec. 5.2 to model cross-tolk between different sensory elements of a compressive device. These results, which are presented in Fig. and Table 7 demonstrate a gentle decrease in reconstruction performance with increase in the cross- talk factor Ç (i.e., increase correlation in the elements of the rows of the sensing matrix). Nonetheless, the overall reconstruction quality remains strong across all images when using the Debiased LASSO.
13

---

Published in Transactions on Machine Learning Research (03/2026)
Original
LASSO
Debiased LASSO
Cov. Prob.
Barbara
Cameraman
Moon
Male
Figure 3: Compressive Image Reconstruction: Reconstruction of grayscale images of dimension 256 x 256 (leftmost column) using LASSO (column 2 from left) and Debiased LASSO (column from left) and the empirical coverage probabilities N./K Vi E [p] based on confidence intervals for the debiased LASSO estimates (rightmost column). The sensing matrix contains random Rademacher entries. Here 20000, p 65, 536 and the additive noise 0.01x the mean absolute value of the noiseless measurements. Debiasing matrix computation using 4 (method from Javanmard & Montanari 2014a) would have taken more than 2 days, whereas our Fast Debiasing approach accomplishes it in less than second.
14

---

Published in Transactions on Machine Learning Research (03/2026)
Computation Time: In both cases above, note that the fast debiasing approach allowed for construction of the debiasing matrix W in about a second. On the other hand, using the method from Javanmard & Montanari 2014a to construct M would have required more than two days.
Original
LASSO
Debiased LASSO
Cov. Prob.
Barbara
Cameraman
Moon
Male
Figure 4: Compressive Image Reconstruction given cross-talk (correlation in the entries of the sensing ma- trix): Reconstruction of grayscale images of dimension 256 x 256 (leftmost. column) using LASSO (col- umn from left) and Debiased LASSO (column from left) and the empirical coverage probabilitics N Vi E [p] based on confidence intervals for the debiased LASSO estimates (rightmost. columm). Here 20000, p 65, 536, and the additive noise is taken as 1% of mean absolute noiseless measurements and 0.02. Debiasing matrix computation using (4) (method from JJavanmard & Montanari (2014a)) would have taken more than 2 days, whereas our Fast Debiasing approach accomplishes it in less than second.
6.2 Experiments on Compressive Video Reconstruction
Here, follow the forward model of well known video compressed sensing architectures such Liu et al. 20133) which acquire snapshot images representing the superposition of set of pixel-wise modulated con- secutive video frames. This snapshot image y E TR?² (an image of size p X p, reshaped to form vector) is
15

---

Published in Transactions on Machine Learning Research (03/2026)
RRMSE
SSIM
Cov. Prob. (Edges)
Cov. Prob. (Non-Edges)
Barbara 0.0445
0.901
0.65
0.90
Cameraman 0.0481
0.909
0.61
0.89
Moon 0.0317
0.923
0.72
0.92
Male
0.0457
0.898
0.58
0.89
Table 7: Reconstruction quality metrics for grayscale images of size 256 x 256 using Debiased LASSO given cross-talk (correlation in the entries of the sensing matrix) with Ç = 0.02. The metrics reported include RRMSE, SSIM, and average coverage probabilities computed separately on edge and non-edge pixel locations basod on the Canny edge detection technique.
represented in the form:
y Σ
2, o f. +n
(9)
where v) E Re² is a noise vector with elements drawn from (0, o²), & R randomly generated Bernoulli or Rademacher pattern for modulating frame ft. the f-th frame of the underlying video. The aim is to reconstruct the video f : (fح² from y and {P:) The video is 3D signal of size p x p T, which after vectorization can be regarded as a vector in RRTp" The effective sensing matrix + (of size p² x Tp²) has the form = (diag( $ (e))ldia(($2))) (22) (i)g()))) where (ia a diagonal matrix of size 1² p² containing the elements of 4, its diagonal.
For our experiments, we represented the 3D signal f E RR-TT in the 3D-DCT basis END in the form f = -= o where O E R is a (weakly) sparse vector of 3D DCT coefficients. The snapshot images were obtained by simulating the forward model on an already available video. The video was reconstructed by the LASSO using == arg ming ly - 22 xl|0111, followed by a debiasing step where the debiasing matrix was computed using our fast approach. Reconstruction results for the first set of frames are shown in Fig. the supplemental material contains comparative reconstructions in video format. These reconstructions reveal good quality reconstruction of spatial textures well as temporal motion patterns.
Here again, the aim of the experiment here is to show that the fast debiasing approach works for another compressive architecture where the sensing matrix 4 consists of a column-wise concatenation of diagonal sub-matrices. The method of JJavanaard & Montanari (2014a) would have taken more than 2 days for computing the debiasing matrix, whereas our technique obtains W in less than second.
6.3 Experiments with Hyperspectral Image Reconstruction
We also experimented with roconstruction of hyperspectral images from real compressive acquisitions in the form of coded snapshot images acquired by the CASSI (Coded aperture snapshot spectral imager) camera Kittle et al. (2100). Consider a hyperspectral image f of size Pa x Py X nn where ng, is the number of spectral channels (or wavelengths). We represent f as vector in IRP-PoRE A hyperspectral image can be regarded stack of waveeenggt--peccfft slices, where each slice is an image of size Pa Py- The CASSI camera does not measure the entire image f, but instead measures a coded snapshot image y (roughly of size Pa. X Py: represented as vector in IRP-Py) in the form of a superposition of the L individual slices each modulated by binary patterns. This is expressed mathematically in the following manner:
f. C.
(10)
where f, (a vector in IPePy) is the ith slice of f and C, (a vector in (0,1)PPPv) is the binary code for the ith slice. The binary codes are implemented in hardware via coded aperture (or mask) that modulates the white light entering the camera. A prism inside the camera disperses this light into its constituent wavelengths and also gives rises to different shifts to each wavelength, ensuring that each slice is modulated
16

---

Published in Transactions on Machine Learning Research (03/2026)
Frame
Frame 2
Frame 3
Frame 4
Frame 5
Original
LASSO
Debiased LASSO
Figure 5: Video reconstruction for a water fountain scene: Reconstruction results showing the progression from original images via LASSO and Debiased LASSO. Each column corresponds to an individual frame and the first row corresponds to 5 frames of the original video, the second row shows the frames reconstructed using LASSO, the third row shows the frames reconstructed using Debiased LASSO. The compressed mea- surements correspond to T= 5 in (9). The average RRMSE and SSIM over the 5 frames are given as 0.0189 and 0.9293 respectively. Debiasing matrix computation using 4 (method from Javanmard & Montanari (201 4a)) would have taken more than 2 days, whereas Fast Debiasing accomplishes it in less than a second. See supplemental video for results in video format.
17

---

Published in Transactions on Machine Learning Research (03/2026)
Figure 6: Coded snapshot used for hyperspectral reconstruction in Fig.
by a different binary code before the superposition of the individual slices recorded by the sensor. For more details, see Kittle al. (2010). The aim to reconstruct f from y and {C,) Since the associated compression ratio here (nz 1) very high, in practice a multi-snapshot version of CASSI is used, where T nt different coded snapshot images acquired, each with different coded aperture pattern. The forward model now is:
Vt E T) y: Σ f. Ca n.
(11)
where y. is the 1-th coded snapshot, Ca (a vector in {0, 12 ) is the coded aperture pattern for the ith channel in the t-th snapshot. The aim to reconstruct f from (ye and ( Here again, we employ a LASSO estimator with 3D-DCT representation for f = ve where 3D-DCT basis matrix and 0 IR is vector of 3D-DCT coefficients. The estimate of 0 obtained by minimizing | T x| where z is vector of size TpaPy 1 obtained by concatenating all the vectorized coded snapshots (1,) and + is a matrix of size TPaPy NLPaly defined follows:
xp=PynexxaPppt is the
y1
diag(Cr.)) diag(C2.1)
diag(C. C
(12)
UT
diag (CTr) diag( C2.) diag(Cnn,.)) Cn..r))
where Vi E [nz]. Vt E [T], diag( C.,)) denotes diagonal matrix of size PxPy x PaPy containing the PaPy elements of Ct., along its diagonal. For our experiments, we used a set of = coded snapshot images acquired by a real camera, corresponding hyperspectral image of size 1021 730 24 with nı 24. One of these snapshot images is shown Fig. G Four different slices of the reconstructed hypercube using LASSO and debiased LASSO (using our fast approach for computing the debiasing matrix) are shown in Fig. 7 Since ground truth is absent, RRMSE SSIM values cannot be directly computed. However the reconstruction with T = (compression ratio of 4:1) snapshots can be compared to a reconstruction with T = 24 snapshots (compression ratio 1:1). The latter can be regarded as a form of ground truth since there no compression. We observe that the debiased reconstructions with T snapshots quite closely match those with T = 24, validating the success of our debiasing approach on this architecture as well. The RRMSE and SSIM values between reconst.ructins under T and T' 24 averaged over all bands are 0.0023 and 0.9912 for LASSO and 0.0029 and 0.9891 for Debiased LASSO rospectively.
7 Conclusion
In this article, we reformulate the optimization problem to obtain M (the approximate inverse of the covari- ance matrix of the rows of the sensing matrix A) in Javanmard & Montanari (2014a) and further provide
18

---

Published in Transactions on Machine Learning Research (03/2026)
Lasso (1:4)
Deb. Lasso (1:4)
Lasso (1:1)
Deb. Lasso (1:1)
(426. nm)
Band 3
Band 7 (478.3 nm)
Band 11 (530.4 mm)
Band 15 (582.6 6 mm)
Band 19 (634.8 nm)
Band 23 (687.0 nm)
Figure 7: Reconstruction of a hyperspectral image of size 1021 x 730 x 24: Rows correspond to bands 3, 7, 11, 15, 19, and 23 (wavelengths 426.1 687.0 nm). Columns correspond to reconstructed images using LASSO and Debiased LASSO reconstructions under 1:4 (that is, with T 6 different coded snapshots in (11) and 1:1 (that is, with 24 different coded snapshots in (IIb) compression ratios respectively. The slices any wavelength are represented using the color corresponding to the wavelength. The RRMSE and SSIM values between reconstructions under 1:4 and 1:1 averaged over all bands are 0.0023 and 0.9912 for LASSO and 0.0029 and 0.9891 for Debiased LASSO respectively.
exact, closed-form optimal solution to the reformulated problem under assumptions the pairwise inner products of the columns of A. For sensing matrices with i.i.d. zero-mean sub-Gaussian rows that have a diagonal covariance matrix or a full covariance matrix with small-valued off-diagonal elements, the debiased LASSO estimator, based on this closed-form solution, has entries that are asymptotically zero-mean and sub- Gaussian. The exact solution significantly improves the time efficiency for debiasing the LASSO estimator, as shown in the numerical results. Our method particularly useful for debiasing in streaming settings where measurements new signal features arrive on the fly.
19

---

Published in Transactions on Machine Learning Research (03/2026)
References
Michal Aharon, Michael Elad, and Alfred Bruckstein. K-SVD:
An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on signal processing, 54(11):4311 4322, 2006.
Jonathan M. Borwein and Adrian S. Lewis. Convex Analysis and Nonlinear Optimization: Theory and Ezamples. CMS Books in Mathematics. Springer, 2nd edition, 2006. ISBN 978-0-387-81256-3, doi:
Stephen Boyd and Lieven Vandenberghe. Conver optimization. Cambridge University Press, 2004.
E. J. Candès, J. K. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measure- ments. Comm. Pure Appl. Math., 50(8)):207-1223, 2006. doi: https://doi.orr/10.. 10110000///2a/211///2
Marco F Duarte, Mark A Davenport, Dharmpal Takhar, Jason N Laska, Ting Sun, Kevin F Kelly, and Richard Baraniuk. Single-pixel imaging via compressive sampling. IEEE signal processing magazine, 25(2);83-91, 2008.
Simon Foucart and Holger Rauhut. A mathematical introduction to compressive sensing. Springer, 2013.
Trevor Hastic, Ryan Tibshirani, and Martin Wainwright. Statistical Learning with Sparsity: The LASSO and Generalizations. CRC Press, 2015.
Frederik Hoppe, Claudio Mayrink Verdun, Hannah Laus, Felix Krahmer, and Holger Rauhut. Non- asymptotic uncertainty quantification in high-dimensional learning. Advances Neural Information Pro- cessing Systems, 37:122524 122555, 2024a.
Frederik Hoppe, Claudio Mayrink Verdun, Hannah Sophie Laus, Sebastian Endt, Marion Irene Menzel, Felix Krahmer, and Holger Ranhut. Imaging with confidence: Uncertainty quantification for high-dimensional undersampled MR. images. In European Conference on Computer Vision, pp. 432 092-450,-4 2024b.
Adel Javanmard and Andrea Montanari, Confidence intervals and hypothesis testing for high nigh-dimensional regression. J Mach Learn Res, 2014a.
Adel Javanmard and Andrea Montanari. Hypothesis testing in high-dimensional regression under the Gaus- sian random design model: Asymptooic theory. IEEE Transactions Information Theory, 60(10):6522 6554, 2014b. doi: 10.1109/TITI 2..109//TTT.2014.23333 2.14.23662..
David Kittle, Kerkil Choi, Ashwin Wagadarikar, and David Brady. Multiframe image estimation for coded aperture snapshot spectral imagers. Applied optics, 49(36)66824 6833, 2010.
Kuldeep Kulkarni, Suhas Lohit, Pavan Turaga, Ronan Kerviche, and Amit Ashok. Reconnet: Non-iterative
reconstruction of images from compressively sensed measurements. In Proceedings of the IEEE conference on computer vision and pottern recognition, pp. 449 009-458,44 2016.
Gitta Kutyniok and Demetrio Labate. Shearlets: Multiscale analysis for multivoriate data. Springer Science & Business Media, 2012.
Sai Li. Dobiasing the debiased LASSO with bootstrap. Electronic Journal of Statistics, 14, 2020.
Dengyu Liu, Jinwei Gu, Yasunobu Hitomi, Mohit Gupta, Tomoo Mitsunaga, and Shree K Nayar. Efficient space-time sampling with pixel-wise coded exposure for high-speed imaging. IEEE transactions pattern analysis and machine intelligence, 36(2):248-200) 2013.
Sara Van de Geer, Peter Bühhlann, Vo'acov Ritov, and Ruben Dezeure. On asymptotically optimal confi- dence regions and tests for high-dimennional models. The Annols of Statistics, 42(3):1166 1202, 2014.
Omar Vazquez and Bin Nan. Debiased LASSO after sample splitting for estimation and inference in high- dimensional generalized linear models. Canadian Journal of Statistics, 53(1 (3(1)=el11) (ele)ell12))) 1827, 2025.
20

---

Published in Transactions on Machine Learning Research (03/2026)
Roman Vershynin. High-Dimensional Probability.An Introduction with Applications in Data Science. Cam- bridge University Press, 2018.
Lu Xia, Bin Nan, and Yi Li. Debiased LASSO for generalized linear models with a diverging number of covariates. Biometrics, 79(1):344 -577, 2023.
Yibo Yan, Xiaozhou Wang, and Riquan Zhang. Confidence intervals and hypothesis testing for high- dimensional quantile regression: Convolution smoothing and debiasing. Journal of Machine Learning Research, 24(245):1 199, 2023.
Cun-Hui Zhang and Stephanie S. Zhang. Confidence intervals for low-dimenssinal parameters in high- dimensional linear models. Journal of the Royal Statisticol Society: Series B (Statistical Methodology), 76 (1):217-242, 2014.
A Appendix for 'Fast Debiasing of the Lasso Estimator': Proofs of Theoretical Results
A.1 Proof of Theorem 1
Primal feasibility: If lffe S μ then we have that μ - up 2 p which implies that S (1 u)p S n.
The choice of w. given by is primal feasible since
1 A
a e,
S max |(1 p)pp = μ.
(13)
n
To see why this true, note that for index j, the LHS is upper bounded by μ4, otherwise it is upper bounded by [11 - p)pl.
Primal objective function value: The primal objective function value is given by 1 51w,112
(1 (1-m)² a 1l,ll//n (1 1)
([a,1B/n)2
| 112/m
The Fenchel dual problem: Consider optimization problem of the form for fixed [p]:
1
inf f(w) + gj
A
(14)
where ff and 9j are extended real-valued convex functions. The Fenchel dual (sce Chapter 3 of Borwein & Lewis 2006)) is
sup (=A (-
(15)
where f* and 97 are the convex conjugates of and 9j respectively. The Fenchel dual satisfics weak duality (see Chapter of Borwein & Lewis (2006)) ) i.e., for any w and u,
1 - w
2 -5" (H (-v)
f(w) gi
In our setting, for fixed j, we consider
l| es S r
f(w) == | and ()(w)= =
(16)
8
otherwise
Then, for the same j, we have their convex conjugates from Lemma
f*(u) = sup u T
f(w) 4 ||ul1|
(17)
s0
(7) = sup u T
g)(w)
sup u w uy + #1/u//11
(18)
w
|| e, I SH
21

---

Published in Transactions on Machine Learning Research (03/2026)
This gives dual problem in the form sup& in u A Au + U 11/2|111
The point u = 2(1 - nee is feasible for the dual (trivially, there constraints).
2(1 p0e ||a.5 12/n
Dual objective function value: Plugging in u
the corresponding dual objective function
value is
1 u A Au + uj -n||u|l1 4n
1
4(1 12)
2(1 11)
2(1 14)
|,,
4n
(1 l12/n))
[a.j 1///
a.. 1
(1 -1- =1)²
(1 un)
(1 p))
|a.
||.3] 11///
Since the primal solution and the dual objective function values are equal, it follows that an optimal solution for the primal is
|a.j 1221///" a.j and that an optimal solution to the dual is 2(1 11)
(1 14)
||. Ea.
We have shown that if p/(1 + p) S S then the optimal solution of 6 is given by 7 Now consider the case when < p1+ This implies μ < (1-1)p. Let i,j E [p] (with o j) be such that = a |/,
Then plugging in the expression w. ||. 11
a(1-H) a.j from into the constraint of have,
-
a 11
a.j
2 (1 a (1 --)pp>>
This shows that w (defined in (7) is not feasible for when μ < p1+ and so is certainly not optimal.
Finally, consider the case when μ 1. If μ 1, then the unique optimal solution of 66) w.j This is because is feasible and is the global minimizer of the objective function. However, when μ >> 1, the
formula 7 does not give the value 0, and so is not the optimal solution to (6)
This concludes the proof that e_ S S 1 is necessary and sufficient condition for the expression given in to be optimal.
A.2 Proof of Theorem 2
For an n X malrix A, let for all j E lp),
1
Lj = -
aa l1
(19)
n
and let for all * E [p],
My == a T a
(20)
Using union bound
32 of Lemma and 31 of Lemma
have under the assumption >
802 a logp for all 1 1 jje [p).
02 Cm
Vlj
2 2C
2m
/
logg +
3
P
<
(21)
L
cEij
D
Given the definition of in Theorem have the bound
a a
Mij Lj
Sp = max
MAX
(22)
+
|a
[
22

---

Published in Transactions on Machine Learning Research (03/2026)
Taking union bound over 1 + j E [p], we have,
2V20 maxh2
logg
p
Vii
Egyl Engs
p(p
1)
S max
<
max [F
+
max 1Fj
2
(23)
1 1+1 [Fj
Li
2
p
2p
logp
2v20 maxx
log
2v 2020maach
Since Egj 2 Cmin for all E [p], we have, maxtzj
Furthermore,
cCann
Cmin
Eigl
given 7
62 Cmax
logg
maxtyj
Enj
we have,
2² Omax
logp
3
P
1+p (2V2+
2 1
c Cuin
2p²
We have now established the upper bound p/(1 + p) with high probability. Theorem states that for
p1 + p) < 1 the optimization problem in 6 is feasible and the optimal debiasing matrix W in (6)
is given by
The choice μ (2v/ 7) Cuaz Cmmn
logg with c (1v2+7) 1) ensures that μ < and
(2/2tz
n
p + p) S μ with high probability.
This completes the proof of Theorem 2
A.3 Lower bound Lij
In Lemma we show that for ensemble of sensing matrices satisfying assumptions D1, D2, the parameter Ly is greater than e Ejj for all j E [p), with high probability, for some constant c.
Lemma 1 Let A be x matrix with independently and identically distributed Sub-Gaussian rows, where n < p. Consider L as defined in (19 For any constant c E (0, 1) and k := ||E~1/2a,, 10..llls: if A satisfies properlies D1 ond D2 and n 2
[1- logp, then for all j E [p],
2
P(L 2 20
(24)
p²
Proof of Lemma We have for all j E [p),
| 1 Σ
at.. Since the a, (for i E [n]) are sub-Gaussian, || llue- By the definition of the sub-Gaussian norm
the aıj sub-Gaussian for each j [p] and ||a.s ll
(see footnote in Sec. 2 with g = 2), we know that
マ Elayyl < 1l = |le] a a S l
(25)
Recall that = E 1/2 a, ll in property D1 of sensing matrix A. We have
||ai1lve sup
| 1/2
T
1122
1/2 a:
S
1/2 |//20/12
1
(2/22)) Tx- 1112 1/2 aj.
sup vES
1/2 /2/22
Σ 1/2 1112 -
(1/222) 2) T 0:
sup
sup
||>//2/|10
-ES
Tmax E ///1/1
1/2
11
Vmmaxh K,
(26)
23

---

Published in Transactions on Machine Learning Research (03/2026)
S 22mmxxh2 From the where Cmax is defined in property D2. Therefore, we obtain Elatyl 1 22
definition of cigenvalues, for any æ E IR,,, x: Ex: 2 min [=]|= 11 2 mminl|=]13 Putting x ej. where e, is
the jth column of I, we have, Σn 2 Ciin- Since E[a زز 2 Ciin. we have, E | E diy) 2 Cmin-
For a given j E [p], the variables al are independent for all i E [n]. Hence, using the concentration inequality of Theorem 3.1.1 and Equation (3.3) of Vershynin (2018). we have for t
P ( (11aa Ellaa F 112/mll 2) S 2e
20 muxx"
(27)
Using the left-sided inequality of 27 we have,
P ((Ia (1a 1/1la,1///0 - 1) 2e
2e mo
(28)
Using , = Σزj 28 can be rewritten as follows for t O:
PL, S Eg - t) S 2e 2c
(29)
Putting = 20xtk V logg in 29 we obtain:
Cmax 2 Σnj
log P
2
11 Eزك
1 - 2
(30)
n
For some constant E (0, if
8C
80 1= [1-c
logp 2
8C2 M 11-
x logp for all j E [pl, then 30 becomes:
min
2 pd
2
P(Ls ScE 5
P(L 2 cEii)
(31)
p
This completes the proof.
A.4 Upper bound on M7)
In the upcoming Lemma we provide a high probability upper bound on Mi A4 A j E [p], for sensing matrices with independent and identically distributed zero-mean nub-Gaussian rows.
Lemma 2 Let A be a n X dimensional matria salisfying assumptions D1 and D2 and with sub- Saab-Gasssin 23-1/2ag. //2ai.loo. Define vnj as in 20 Then for all + jEjjl [p],
norm K = |
logg + {5sl) 2 1
P ml S 22220mmac maxk?
(32)
n Proof of Lemma 2 We have - - a.j Here, for given I, we know that aay and ag
are independent zero-mean sub-Gaussian random variables. From from (25) and (66 know that their sub-Gaussian norm is at most VCnaxt for all i E [n]. Using Lemma 2.7.7 of Vershynin (2018), we have that for all i E [n], 0iy0a are independent nub-Exponential random variables with sub-exponential norm at most 2 Moreover, Ela,jau] = E₁t- Hence, using Bernstein's inequality for averages of independent sub-cxponential random variables, given in Corollary 2.8.3 of Vershynin (2018), we have for any > 0,
M E Blagau] 2( (p)))r(())
n
Cijaa
agay - Σig 2t S 2e
2012
(33)
F1
T1
We have 1/2, and 2/ in Equation (3.3) and the equation immediately preceding in Vershynin 2018
24

---

Published in Transactions on Machine Learning Research (03/2026)
Using Reverse Triangle's inequality, we have,
1
1 aiaa - ΣZg
ast
- أزاز I
1=1
Thereofore, (1 E aigaal
{ Ei=1 aijai
Eiglh J. Hence, we have,
(
p
asj
2 + |in
$ 2e zetham
(34)
Taking t 2V20maxhi
logp we have for all + j E [p]. V
logg
P Vig 2 2v 2V 2020max
max x22 V
+ |Σnl s pd
(35)
n
This completes the proof.
A.5 Convex conjugates
The convex conjugate of a function f(w) is defined as:
f (w) sup (u T - f(w)).
(36)
The following result gives the convex conjugates of the functions needed in the proof of Theorem
Lemma 3 1. If f(w) = 1 then its conver conjugate is ("( = 11
2. 1f 93 is the indicator function of the convex set (w E RPP 1| -e--ell lm S sJ) ie.
()(w) = 0
if w w SP otherwise,
x
then its convex conjugate is (u) = uj 11 a/|2l|i.
Proof of Lemma
1. We can write f(w)= w Qw where Example 3.2.2 of Boyd & Vandenborghe (0044) form is
Q 2 LP is positive definite (and has size p). From the convex conjugate of a positive definite quadratic
('()) = 2" Q
1
2 n
-1 u
uul3.
u
2 u
gj is the indicator function of the set C, the convex conjugate is given by
g7( = sup T w, wec
(37)
where C = {w E IP || - ei1oo I μ}- This implics that w; E [e 14, Cja + M4, Vi. (Note that
Ci = 1 if = j and 0 otherwise.) To maximize u T w En u,w, the optimal we can be chosen as
es + p if us 2 0,
W0: =
(38)
Cji p if O.
Substitating into u Tww,
w, we obtain u w Ei= up (e + usign(us)), where sign(u,) is the sign of ue Simplifying, we have u TTw w = uj + 1 Σ - |u:]. Thus, we have
0 (u) uj + /1//2|11.
(39)
This completes the proof.
25


Reviews:
[{"review_id": "oz6UwSOLpx", "reviewer": "Reviewer_sNBF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the computational bottleneck of the debiased LASSO estimator, specifically the expensive optimization required to find an approximate inverse of the covariance matrix. By re-parameterizing the problem to focus on a weight matrix $W$ (where $W=AM^\\top$), the authors derive a simple, closed-form solution that eliminates the need for iterative solvers, which holds when the linear correlation satisfies a condition. They provide theoretical guarantees showing that this closed-form solution is optimal under a condition on the sensing matrix's column correlations, which is satisfied with high probability for common random designs. Empirical results demonstrate that this approach maintains statistical inference quality while speeding up computation by several orders of magnitude.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors provide solid proofs using Fenchel duality and concentration inequalities to support both the optimality and the high-probability feasibility of their solution.", "audience_yn": "Yes", "audience_explanation": "Researchers and practitioners in the statistics community would find this paper highly relevant, particularly those focused on high-dimensional statistics, optimization, and scalable machine learning.", "requested_changes": "The reviewer is not an expert on the LASSO estimator and hence can not provide a holistic evaluation of its value. However, the reviewer believes that the proposed method is interesting in itself and the analysis is basically correct. Here are some questions or comments for the paper:\n1. Is it possible to provide a draft on how to derive the closed-form (7) in the main body?\n2. Does Theorem 1 suggest using $\\mu=\\frac{\\rho}{1+\\rho}$ so that the proposed method consistently achieves an optimal error $\\mu$?\n3. It is better to add a set of experiments to compare the proposed method and the baseline in terms of both accuracy and time under different $\\mu$. It would be helpful to show how we should choose different methods based on different $\\mu$ requirements.", "broader_impact_concerns": "This paper focuses on developing algorithms for LASSO estimation. We do not identify any specific societal risks that need to be highlighted in this context.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "erRhTjjarl", "reviewer": "Reviewer_fhgi", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "\\textbf{Summary of Contributions.}\n\nThis paper presents a reformulation of the classical debiased LASSO approach by Javanmard \\& Montanari (2014a). Instead of computing an approximate inverse $M$ of $A^\\top A/n$ via convex optimization (which is computationally expensive), the authors propose to directly compute the weight matrix $W = AM^\\top$, showing that theoretical guarantees of the debiased estimator depend only on $AM^\\top$, not on $M$ itself.  \n\nThe key contribution is the derivation of a closed-form solution for $W$ under mild conditions on the sensing matrix $A$, particularly when the columns of $A$ are weakly correlated sub-Gaussian random vectors. The authors establish deterministic conditions for optimality (Theorem~1) and probabilistic bounds for random designs (Theorem~2), showing that the proposed solution matches the optimization-based one with high probability.  \n\nEmpirical results demonstrate that the closed-form debiasing yields equivalent estimation accuracy and statistical inference to the original debiased LASSO, but with four orders of magnitude faster runtime. Extensive experiments compare sensitivity, specificity, and variance ratios across Gaussian and correlated covariance structures, consistently showing that the proposed method maintains theoretical validity while being computationally efficient.\n\nStrengths:\n\n1 A clean theoretical insight that reduces an iterative convex optimization problem to a closed-form expression.\n2 Strong proofs of equivalence and probabilistic bounds (Theorems~1–2).\n3 Solid empirical validation showing identical performance and major speed-ups.\n4 Practical importance for high-dimensional inference and real-time or streaming regression settings.\n\n\nWeaknesses:\n\\begin{itemize}\n  \\item The assumptions (e.g., sub-Gaussian i.i.d. design, weak correlations) may be restrictive in real-world data.\n  \\item The numerical experiments focus mostly on synthetic setups; no real-data illustration is provided.\n\\end{itemize}", "claims_and_evidence_yn": "Yes", "claims_explanation": "The theoretical claims are well supported through rigorous derivations and clear proofs, including convex duality analysis and probabilistic concentration bounds. The equivalence between the original optimization and the reformulated problem is mathematically sound and verified both analytically and empirically. The experiments convincingly validate the theoretical results—the closed-form solution produces identical debiased estimates to those obtained via optimization, confirming the correctness of the formulation. Runtime analysis further supports the claim of a significant computational advantage.\n\nMinor concerns include the narrow focus on synthetic data and the absence of robustness tests beyond Gaussian or sub-Gaussian ensembles, but these do not undermine the main theoretical claims.", "audience_yn": "Yes", "audience_explanation": "The proposed method directly targets a fundamental limitation of high-dimensional inference—the bias of the LASSO estimator—and provides a computationally efficient debiasing technique. This topic is highly relevant to the machine learning and statistics communities, especially for researchers working on sparse estimation, convex optimization, and scalable inference. Moreover, the result has practical implications for modern high-dimensional applications (e.g., genomics, compressive sensing, causal inference), making it a strong fit for the TMLR readership.", "requested_changes": "**Discussions on choices and estimation of $\\mu$**: The work mentions empirical choices for $\\mu$ but does not give a prescriptive procedure for practitioners. Guidance (or an algorithm) to estimate a safe $\\mu$ from data, taking into account trade-offs, would substantially increase the practical utility.\n\n**Numerical scalability**: Experiments are informative but limited to moderate p (e.g., p=500). Showing scalability to larger \np (and reporting memory/time for both closed-form and optimization) would underline the practical speed advantage.\n\n**Comparisons to other debiasing approaches**: The paper briefly references nodewise regression and other approaches; empirical comparison (runtime and inferential quality) to a representative nodewise-regression implementation would contextualize the gains more concretely.", "broader_impact_concerns": "The paper focuses on methodological and theoretical improvements for statistical estimation. It does not raise ethical or societal concerns. The technique improves computational efficiency for debiased estimators, which may indirectly benefit applications involving large-scale data or scientific inference. No issues of fairness, privacy, or misuse are apparent.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "hRW1EKdvYz", "reviewer": "Reviewer_1iZg", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a computationally efficient reformulation of the classic debiased Lasso framework introduced by Javanmard and Montanari (2014). The key contribution lies in reparameterizing the optimization problem used for constructing the debiasing matrix. The reformulation preserves all theoretical properties of the original debiased Lasso estimator while enabling the derivation of a simple closed-form solution for 𝑊 under a deterministic condition that holds with high probability for many random sub-Gaussian sensing matrices.\n\nStrengths:\n\n1. The reformulation from 𝑀 to 𝑊 elegantly simplifies the debiasing process while preserving theoretical rigor. \n2. This paper provides both theoretical clarity and substantial computational acceleration.\n3. The approach applies to general sub-Gaussian designs and not just to specific covariance structures.\n\nWeaknesses:\n1. The experiments, while detailed, are limited to synthetic data; real-world evaluations or robustness tests against model misspecification are absent.\n2. Selecting an appropriate $\\mu$ in practice may require simulation or prior knowledge of $\\rho$, reducing ease of use in non-ideal settings.", "claims_and_evidence_yn": "Yes", "claims_explanation": "I briefly checked the proofs and did not find any errors.", "audience_yn": "No", "audience_explanation": "This work is too theoretical. I am wondering if any existing works related to this topic. The baseline is from 2014, which is very old and this demonstrates that no new works have been proposed in recent years.", "requested_changes": "The authors could add more related works from recent years to demonstrate that this work is still of interest to a significant group of researchers. \n\nExperiments on real-world data should be included.", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)