{"id":"726afbd9-eb0e-4c3d-86fa-498aa7054a44","arxiv_id":"2412.03269","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A sample-complexity bound for l1-TV compressed sensing is derived, and an unrolled proximal-gradient solver (LPGM-ISTA) recovers ECG signals with better accuracy and speed than standard iterative methods.","lead":"This paper derives a measurement bound for compressed sensing when signals are both sparse and have sparse gradients, using a combination of l1 and total variation penalties, and builds a fast learned solver by unrolling a new iterative method. The theory extends existing guarantees for l1 and TV to the fused-lasso style l1-TV model, and the learned solver recovers ECG signals more accurately and faster than conventional iterative algorithms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1(a) does not prove a single step size works for all iterations; the induction only gives per-iterate u_k with no positive lower bound, so global convergence to a minimizer is unsupported.","rationale":"The reader's conditional verdict is appropriate, but I would anchor it slightly differently. The reader's weakest_assumption is the bounded-domain hypothesis in Theorem 4.2, which is indeed a genuine defect: the theorem's error bound uses r and 2r as if g1(x)=λ1||x||1 had a finite domain, which it does not, so the parameter-selection guarantee is not proved for problem (1.5). I agree with that. However, I think the more load-bearing gap is Theorem 4.1(a): the claimed global convergence to a minimizer is the foundation for the algorithm and for the unrolled solver's initialization, and the proof's induction only shows existence of a step size u_k for each finite prefix, with no uniform positive lower bound as k grows. This is precisely the kind of missing support that should block acceptance of the algorithmic claims as stated. The sample-complexity theorem, by contrast, appears to survive scrutiny: the descent-cone estimate follows a standard Gaussian-width route and the reduction to [52, Corollary 3.5] is coherent, apart from a typo in Section 6.1 where |Sc_R| is written as sr-1 and the false comparison in Remark 1. The false Remark 1 is embarrassing but not load-bearing for Theorem 3.2. Thus the reader's CONDITIONAL verdict stands; my concern reinforces it rather than moving it to another category.","tokens_in":28809,"tokens_out":20165,"duration_ms":182273,"concrete_test":"Take n=2, m=1, A=[1,0], y=0, λ1=λ2=1, x0=(1,1). For any candidate u>0 and t∈(0,u], run PGM-ISTA and at each iterate compute M_k = max_{i: x_{k,i}≠0} (λ1 sign(x_{k,i}) + [A^T(Ax_k-y)]_i)/x_{k,i}, the quantity governed by Lemma 6.3's condition (6.11). If sup_k M_k = ∞ while the iterates remain bounded, no single u>0 can satisfy G_{1/u}^{f,g1}(x_k) ∈ ∂F_1(x_k) for all k, so the proof of Theorem 4.1(a) collapses. An independent re-derivation of the induction with explicit lower bounds on u_k would settle whether a uniform u exists.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.1(a) claims that there exist u>0 and t>0 such that PGM-ISTA converges to a minimizer of (1.5). The proof rests on Lemma 6.3 and the claim (6.18) that G_{1/u}^{f,g1}(x_k) ∈ ∂F_1(x_k) for every k with one common u. But the induction proves only that for each finite k there is some u_k>0, via Lemma 6.3's condition (6.11) whose denominator contains x_i. After choosing u=min{u0,u1}, the next step says 'we can always find u>0' for x_{k+1}; nothing shows these u_k are bounded below away from 0, and if some nonzero coordinate x_i→0 as k→∞, the required 1/u can diverge. Consequently, the identification of Algorithm 1 with the proximal subgradient method for the entire infinite sequence is not established. The sample-complexity part (Theorems 3.1–3.2) is internally consistent, although Remark 1's comparison with [52] is false: for n=10^5, sr=5000, (3.4) gives roughly 9.94×10^4, larger than the roughly 4×10^4 from [52], so the claimed sharpness for the ℓ1 case is wrong. Still, the missing uniform lower bound on admissible step sizes is the load-bearing flaw for the paper's algorithmic claims.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the l1-TV regularized model (1.4)/(1.5) in compressed sensing. Its theoretical contribution is a sample-complexity bound for l1-TV recovery: Theorem 3.1 bounds the statistical dimension of the descent cone in terms of the sparsity level sr and gradient-sparsity level sg, and Theorem 3.2 converts this into an explicit measurement bound with a Gaussian-width probability estimate. The algorithmic contribution is PGM-ISTA, a proximal-gradient-mapping iteration for (1.5), for which the authors claim global convergence and a parameter-selection criterion, and whose unrolled version LPGM-ISTA is trained on ECG data. The paper reports numerical comparisons of l1, TV, and l1-TV and of LPGM-ISTA against iterative solvers.","tokens_in":29062,"tokens_out":12337,"duration_ms":109155,"significance":"The sample-complexity result is significant if correct: it gives a quantitative recovery guarantee for a combined l1+TV penalty that depends on both signal sparsity and gradient sparsity and reduces, in the limiting cases, to previously known TV and l1 bounds. The derivation is substantial and does not appear to be circular: no constant is fitted, and the Gaussian-width machinery is used in the standard way. The learned solver is also potentially useful, and the authors provide code and experiments on ECG signals. However, the algorithmic theory has load-bearing gaps: the proof of global convergence constructs only iteration-dependent step sizes, and the parameter-selection theorem uses a bounded-domain assumption that is false for the actual l1 term. These issues must be repaired before the algorithmic claims can be accepted.","major_comments":[{"comment":"The proof of Theorem 4.1(a) does not establish the existence of a single pair (u,t) valid for the whole infinite sequence. Lemma 6.3, condition (6.11), gives a u0 depending on the current point x, with the denominator containing the nonzero coordinates xi. The induction in the proof of (6.18) only shows that for each finite k one can find a new u_k after x_{k+1} has been generated; nothing bounds these u_k away from zero. If a coordinate xi tends to 0 as k increases, the required 1/u can diverge, so the identification of Algorithm 1 with the proximal subgradient method on the entire trajectory is not justified. Consequently the claim that the sequence converges to a minimizer of (1.5) is unsupported, and the later statement S*∩F≠∅ (just after (4.5)) is also not proved. The proof also has a circular flavor: x_{k+1} depends on u, but u_{k+1} is chosen after x_{k+1} is observed.","section":"Section 6.3, proof of Theorem 4.1(a)"},{"comment":"Theorem 4.2 as stated does not apply to problem (1.5). The theorem's statement does not include the assumption dom(g1)⊂{x:||x||2≤r}; that assumption is only mentioned in the introductory paragraph before the theorem, and for the actual choice g1(x)=λ1||x||1 the domain is all of Rn, so no finite r exists. The proof uses r in several essential places: inequality (4.6), Lemma 6.4, the estimate ||xk−x*u||2≤2r, and the final bound 2rε+(tκ+β)u. Since r is not defined by the problem data and the boundedness condition fails, the claimed parameter-selection guarantee is unsupported. This is a load-bearing issue because parameter selection is one of the paper's central algorithmic contributions.","section":"Theorem 4.2 and Section 6.4"},{"comment":"The numerical comparison in Remark 1 is incorrect. For n=10^5 and sr=5000, the right-hand side of (3.4) equals approximately 4.25×10^4, whereas the bound 2sr log(n/sr)+2sr from [52] equals approximately 4.00×10^4. Thus (3.4) is not a sharper upper bound in this regime, and the statement that (3.4) gives a tighter bound than [52] for sr<5000 is false. The same comparison is repeated in Remark 4 and in the contribution summary, so the claimed improvement over [52] for the l1 case is not supported. This does not invalidate Theorem 3.2 itself, but it requires correction of the stated comparison.","section":"Remark 1 and Remark 4"}],"minor_comments":[{"comment":"The reduction statement is swapped: if λ1=0 and λ2>0, (1.4) reduces to the TV problem (1.3), not the l1 problem (1.2); and if λ1>0 and λ2=0, it reduces to the l1 problem, not TV.","section":"Introduction, after Eq. (1.4)"},{"comment":"The displayed error bound is missing parentheses: it should read ||x*−x^||2 ≤ 2ε/(√(m−1)−√Φ(sr,sg)−t). As printed, the denominator is ambiguous.","section":"Theorem 3.2"},{"comment":"In the sentence before (6.8), the text states |Sc_R| = sr−1; the correct value is n−sr. The subsequent algebra correctly uses n−sr, so this is a typo, but it should be fixed.","section":"Proof of Theorem 3.1"},{"comment":"The first term after the inequality labeled (c) appears to contain an extra superscript 2 on the norm of (I_n−uA^TA)(x_k−x*); the subsequent bound uses the norm without squaring. Please clarify the displayed formula.","section":"Eq. (6.20)"},{"comment":"The parameter set ΘL is stated as {Wx, Wy, u, t}, but Wx = I_n−uA^TA is determined by u when A is fixed. The redundancy or intentional decoupling of Wx and u should be explained.","section":"Section 4.2, learnable parameters"},{"comment":"In Step 5, the dual update uses µ without a subscript: v^{k+1}=v^k+µ(x^{k+1}−z^{k+1}). This should presumably be µ_k or µ_{k+1} consistently with the other updates.","section":"Algorithm 2"}],"recommendation":"major_revision","confidential_remarks":"The sample-complexity part of the paper appears to be a solid contribution and may be publishable once the algorithmic theory is repaired or explicitly separated from the recovery theory. The issues with Theorem 4.1(a) and Theorem 4.2 are not merely cosmetic: the global convergence proof lacks a uniform step size, and the parameter-selection theorem relies on a bounded-domain assumption that is false for the l1 norm. The false comparison with Tropp's bound in Remark 1 should also be corrected. I would not reject the manuscript outright, because the recovery guarantee and the numerical experiments retain value, but the algorithmic claims need substantial revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The sample-complexity result is the genuine contribution. Theorem 3.2 gives an explicit Gaussian-width bound depending on both sparsity sr and gradient sparsity sg, and it reduces correctly to the known TV and l1 bounds in the appropriate limits. That is new, parameter-free, and follows the established framework without fitting anything to data. The synthetic experiments support the bound reasonably well, comparing favorably against Genzel et al.\n\nThe algorithm theory is where the problems sit. Theorem 4.1(a) claims global convergence to a minimizer, but the proof only constructs, for each finite iterate, a step size u_k satisfying Lemma 6.3. The induction then chooses u = min{u0, u1}, but nothing prevents the required u_k from shrinking to zero as coordinates approach zero. Without a uniform positive lower bound on u, the identification of Algorithm 1 with the proximal subgradient method for the whole infinite sequence is not established. That is load-bearing, not cosmetic.\n\nTheorem 4.2 has a related but more concrete defect: it assumes dom(g1) is bounded by radius r, but the actual g1 = lambda1 * ||x||_1 has unbounded domain, and r is never defined in the theorem statement. The parameter-selection bound therefore does not apply to problem (1.5) as stated. This is fixable, but it has to be stated honestly.\n\nRemark 1 overclaims. For n = 1e5 and sr = 5000, the paper's bound (3.4) is roughly 9.94e4, while the Tropp bound cited is roughly 4e4, so the claim that (3.4) is sharper for sr < 5000 is wrong at the endpoint and needs a corrected range or a different comparison.\n\nThe learned solver experiments are encouraging but limited: LPGM-ISTA with few layers outperforms the iterative baselines by a wide margin on ECG data, which is a meaningful empirical result. But the training details are incomplete, there is no variance or repeated-trial information, and the evaluation is in-distribution only. The paper itself acknowledges the out-of-distribution limitation, which I credit.\n\nWho is this for? People working on compressed sensing with structured sparsity, and practitioners using fused lasso or l1-TV for ECG or similar signals. The sample-complexity core deserves serious refereeing despite the algorithm-theory problems. The right outcome is a major revision: fix the convergence proof, remove or explicitly bound the domain assumption, correct Remark 1, and add variance to the tables. I would send this to peer review, not desk-reject it.","headline":"The explicit sample-complexity bound for l1-TV is a real contribution; the PGM-ISTA convergence theory has a load-bearing gap, and one comparison remark overclaims against Tropp.","tokens_in":29686,"tokens_out":1577,"would_cite":true,"duration_ms":18305,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A12","90C25","65K05","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that $\\ell^1$-TV regularization recovers signals from Gaussian measurements once the sample count exceeds $(\\sqrt{\\Phi(s_r,s_g)}+t)^2+1$, with $\\Phi$ depending jointly on sparsity and gradient sparsity, and unrolls a…","keywords":["ℓ1-TV regularization","compressed sensing","sample complexity","total variation","proximal gradient mapping","algorithm unrolling","learned solver","ECG signal recovery"],"falsifier":"Run the exact setting of Theorem 3.2 for fixed $n$, $s_r$, $s_g$, $\\lambda_1=\\lambda_2=1$ and Gaussian $A$: compute $\\Phi(s_r,s_g)$, take $m=\\lfloor(\\sqrt{\\Phi(s_r,s_g)}+t)^2+2\\rfloor$, and record the empirical success frequency over many draws; if it falls clearly below $1-e^{-t^2/2}$, the sample-complexity bound is false. Separately, compare $F(x_{k+1})-F(x^*)$ with the claimed bound $2r\\epsilon+(t\\kappa+\\beta)u$ on a sequence of iterates with growing $\\|x_k\\|_2$; any finite $r$ violated by the curve would show the hidden bounded-domain assumption is doing real work.","tokens_in":28504,"feed_emoji":"📈","tokens_out":17160,"duration_ms":131572,"temperature":0.7,"pith_summary":"The paper's aim is to make the combined $\\ell^1$-plus-total-variation penalty a theoretically grounded and practically fast way to recover signals that are both sparse and piecewise-constant. Its central theoretical result is a measurement bound: under Gaussian sensing, a signal with $s_r$ nonzero entries and $s_g$ nonzero adjacent differences is stably recovered once the number of measurements exceeds $(\\sqrt{\\Phi(s_r,s_g)}+t)^2+1$, with an explicit formula for $\\Phi$. The same bound reduces to known TV and $\\ell^1$ cases when one penalty is switched off, and in suitable parameter regimes it is smaller than either, meaning the joint model can need fewer measurements. On the algorithmic side, the paper constructs PGM-ISTA, proves global convergence, and unrolls it into LPGM-ISTA, which recovers ECG signals accurately in a handful of layers and far less time than standard iterative solvers.","feed_headline":"ℓ1-TV sample complexity now provably depends on both sparsities","feed_subtitle":"A new bound couples regular sparsity and gradient sparsity, and an unrolled PGM-ISTA recovers ECG signals in milliseconds.","key_machinery":"The load-bearing object is the descent cone $D(g,x)$ of the composite penalty $g(x)=\\lambda_1\\|x\\|_1+\\lambda_2\\|Dx\\|_1$ and its statistical dimension, the expected squared distance from a standard Gaussian vector to the polar cone. To bound the statistical dimension the authors choose vectors $z_1\\in\\partial\\|x\\|_1$ and $z_2\\in\\partial\\|Dx\\|_1$, form $\\lambda_1z_1+\\lambda_2D^{\\top}z_2$, and use a sign-correlation expectation identity to evaluate $\\mathbb{E}\\|g-t(\\lambda_1z_1+\\lambda_2D^{\\top}z_2)\\|_2^2$; minimizing over $t$ yields $\\Phi(s_r,s_g)$, and the inequality $w^2\\le\\delta$ converts the width bound into a measurement bound. The algorithmic machinery is the proximal gradient mapping update $x_{k+1}=T_{\\lambda_2t}((1-t/u)x_k+(t/u)S_{\\lambda_1u}(x_k-uA^{\\top}(Ax_k-y)))$, where $S$ is soft thresholding and $T$ is the TV proximal map; unrolling this update with learnable weights $W_x$, $W_y$, $u$, $t$ yields the LPGM-ISTA network.","core_discovery":"On its own terms, the central discovery is Theorem 3.2: if the unknown signal $x^*$ has regular sparsity $s_r=|S_R(x^*)|$ and gradient sparsity $s_g=|S_G(x^*)|$, and $A\\in\\mathbb{R}^{m\\times n}$ has independent standard Gaussian rows with $y=Ax^*+e$ and $\\|e\\|_2\\le\\epsilon$, then the solution $\\hat{x}$ of the constrained problem (1.4) obeys $\\|x^*-\\hat{x}\\|_2\\le 2\\epsilon/(\\sqrt{m-1}-\\sqrt{\\Phi(s_r,s_g)}-t)$ with probability at least $1-e^{-t^2/2}$, provided $m>(\\sqrt{\\Phi(s_r,s_g)}+t)^2+1$. Here $\\Phi(s_r,s_g)$ is the closed-form quantity in (3.2) built from $n$, $\\lambda_1$, $\\lambda_2$, $s_r$ and $s_g$. The paper also claims that setting $\\lambda_1=0$ recovers the TV-only sampling bound, that setting $\\lambda_2=0$ gives an $\\ell^1$ bound that can be sharper than the standard one for modest sparsity levels, and that PGM-ISTA converges globally for $u\\in(0,2/\\|A\\|_2^2)$ and $t\\in(0,u]$, with its unrolled version LPGM-ISTA outperforming iterative baselines on ECG data.","pith_inferences":["Beyond the paper: the same descent-cone technique should transfer to other composite penalties of the form $\\lambda_1\\|x\\|_1+\\lambda_2\\|\\Psi x\\|_1$ whenever the second operator's subdifferential has a tractable sign structure, giving sample-complexity formulas for overlapping-group and analysis sparsity models.","Beyond the paper: Theorem 4.2's hidden bounded-domain assumption means the stated parameter-selection criterion is not yet a theorem for the actual unbounded $\\ell^1$ objective; a bounded-domain variant or an explicit bound on $\\|x_k\\|_2$ along the iterates would be needed to restore the guarantee.","Beyond the paper: since LPGM-ISTA is trained on a fixed distribution of ECG windows and Gaussian sensing matrices, its millisecond-level advantage is likely distribution-specific; the paper itself notes out-of-distribution failure, so the practical claim is best read as in-distribution acceleration rather than a replacement for convergent solvers in general.","Beyond the paper: the comparison table suggests the joint bound can be several times smaller than either single-penalty bound when $\\lambda_1/\\lambda_2=1$, implying that for signals like ECG the measurement budget itself could be reduced if the bound is tight; that implication is not tested in the paper."],"forward_implications":["If Theorem 3.2 is correct, the sample complexity of $\\ell^1$-TV recovery is governed by $\\Phi(s_r,s_g)$, so the measurement count decreases with stronger value sparsity or gradient sparsity and interpolates between the $\\ell^1$ and TV regimes as $\\lambda_1/\\lambda_2$ varies.","The bound recovers the TV-only sampling number when $\\lambda_1=0$ and gives an $\\ell^1$ bound when $\\lambda_2=0$; for $n=10^5$ and $s_r<5000$ the paper claims the $\\ell^1$ case is sharper than the standard bound.","PGM-ISTA has a guaranteed convergent regime $u\\in(0,2/\\|A\\|_2^2)$, $t\\in(0,u]$, and the convergence analysis gives an explicit function-value error bound that shrinks as $u$ becomes smaller, at the cost of more iterations.","The unrolled LPGM-ISTA network achieves its best reported ECG recovery with very few layers (relative error $0.056$ at two layers and $0.039$ at ten layers) and computes in about $0.06$ to $0.10$ seconds, which the paper contrasts with iterative solvers needing hundreds or thousands of iterations."],"supporting_citations":[{"why":"Defines statistical dimension and supplies the inequality $w^2\\le\\delta\\le w^2+1$ that links the width estimate to the sampling number.","marker":"[1]"},{"why":"Gives the TV descent-cone bound that Theorem 3.1 extends to the composite penalty and that the $\\lambda_1=0$ special case recovers.","marker":"[17]"},{"why":"Supplies the inequality $\\delta(D(g,x))\\le\\inf_t\\mathbb{E}[\\mathrm{dist}^2(g,t\\partial g(x))]$ used to convert the subdifferential calculation into the $\\Phi$ bound.","marker":"[18]"},{"why":"Gives the conic-recovery error estimate that turns the width bound into the statement of Theorem 3.2, and the standard $\\ell^1$ baseline bound.","marker":"[52]"},{"why":"Provides the sign-correlation expectation identity used to evaluate the Gaussian expectations in the proof of Theorem 3.1.","marker":"[53]"},{"why":"Provides the gradient-mapping, proximal-operator, nonexpansivity, and proximal-gradient inequalities underlying PGM-ISTA and Theorems 4.1 and 4.2.","marker":"[5]"},{"why":"Supplies the weak Jacobian of the TV proximal map that lets the outer TV step of PGM-ISTA be differentiated and unrolled in LPGM-ISTA.","marker":"[14]"},{"why":"Establishes the ISTA-unrolling paradigm that LPGM-ISTA extends to the two-penalty $\\ell^1$-TV objective.","marker":"[26]"},{"why":"Gives the competing analysis-$\\ell^1$ recovery bound used as the comparison baseline in the synthetic sampling-number experiments.","marker":"[24]"},{"why":"Supplies the ECG signal data used in the experiments demonstrating LPGM-ISTA's accuracy and speed.","marker":"[44]"}],"fun_headline_variants":["ℓ1-TV sample complexity depends on both sparsities","Unrolled PGM-ISTA solver speeds ℓ1-TV recovery","ℓ1-TV error bound couples regular and gradient sparsity","Fast unrolled solver recovers ECG signals via ℓ1-TV"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The parameter-selection guarantee for PGM-ISTA silently assumes that the $\\ell^1$ penalty term is only evaluated on vectors of bounded length, even though the actual objective allows vectors of any length, and the radius $r$ appearing in the error bound is never fixed by the problem data.","fun_headline_variants_meta":{"raw":{"variants":["ℓ1-TV sample complexity depends on both sparsities","Unrolled PGM-ISTA solver speeds ℓ1-TV recovery","ℓ1-TV error bound couples regular and gradient sparsity","Fast unrolled solver recovers ECG signals via ℓ1-TV"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000597,"raw_usage":{"total_tokens":2832,"prompt_tokens":1026,"completion_tokens":1806,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":1734}},"tokens_in":642,"tokens_out":1806,"duration_ms":12424,"temperature":1.0,"reasoning_tokens":1734,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:37:03.534289+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the exact setting of Theorem 3.2 for fixed $n$, $s_r$, $s_g$, $\\lambda_1=\\lambda_2=1$ and Gaussian $A$: compute $\\Phi(s_r,s_g)$, take $m=\\lfloor(\\sqrt{\\Phi(s_r,s_g)}+t)^2+2\\rfloor$, and record the empirical success frequency over many draws; if it falls clearly below $1-e^{-t^2/2}$, the sample-complexity bound is false. Separately, compare $F(x_{k+1})-F(x^*)$ with the claimed bound $2r\\epsilon+(t\\kappa+\\beta)u$ on a sequence of iterates with growing $\\|x_k\\|_2$; any finite $r$ violated by the curve would show the hidden bounded-domain assumption is doing real work.","supporting_citations":[{"cited_title":"Amelunxen, M","cited_arxiv_id":null,"evidence_quote":"Defines statistical dimension and supplies the inequality $w^2\\le\\delta\\le w^2+1$ that links the width estimate to the sampling number."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the TV descent-cone bound that Theorem 3.1 extends to the composite penalty and that the $\\lambda_1=0$ special case recovers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the inequality $\\delta(D(g,x))\\le\\inf_t\\mathbb{E}[\\mathrm{dist}^2(g,t\\partial g(x))]$ used to convert the subdifferential calculation into the $\\Phi$ bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the conic-recovery error estimate that turns the width bound into the statement of Theorem 3.2, and the standard $\\ell^1$ baseline bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the gradient-mapping, proximal-operator, nonexpansivity, and proximal-gradient inequalities underlying PGM-ISTA and Theorems 4.1 and 4.2."},{"cited_title":"Cherkaoui, J","cited_arxiv_id":null,"evidence_quote":"Supplies the weak Jacobian of the TV proximal map that lets the outer TV step of PGM-ISTA be differentiated and unrolled in LPGM-ISTA."},{"cited_title":"Gregor and Y","cited_arxiv_id":null,"evidence_quote":"Establishes the ISTA-unrolling paradigm that LPGM-ISTA extends to the two-penalty $\\ell^1$-TV objective."},{"cited_title":"Genzel, G","cited_arxiv_id":null,"evidence_quote":"Gives the competing analysis-$\\ell^1$ recovery bound used as the comparison baseline in the synthetic sampling-number experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the ECG signal data used in the experiments demonstrating LPGM-ISTA's accuracy and speed."}],"review_version":1}