{"id":"fcead6d3-2df4-4ce3-ac41-f0d76d72a03f","arxiv_id":"2504.17874","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"SOFARI-R constructs debiased, asymptotically normal estimators with consistent variance estimates for the latent right singular vectors of a sparse SVD regression coefficient matrix.","lead":"This paper proposes SOFARI-R, a statistical method for estimating and computing confidence intervals for the latent right-side factors of a high-dimensional multi-output regression model, extending the earlier SOFARI framework to the response side. It matters because it adds uncertainty quantification to response selection and to principal-component-style inference in multi-task learning.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Weak-orthogonality results hinge on imported, unproved Lemma 10 from the same authors' companion preprint; Theorem 3's expansion and variance formula are conditional on it.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: Theorem 3 and its variance estimator depend on Lemma 10, an imported theorem from the same authors' unpublished companion paper, with no proof or machine-checked verification in the present manuscript. I agree that this is the main soft spot in the central claim. The strong-orthogonality branch appears internally coherent: Theorem 1's proof decomposes the error into four terms bounded in Lemmas 1-4, and the variance computation in Section A.1 checks out dimensionally and algebraically. I would add two smaller observations that do not change the verdict. First, the displayed hard-thresholding formula in Section 2.2 and Algorithm 2, bmu^t_ij = bmu_ij 1(bmu_ij >= log n / sqrt n), lacks the absolute value that Lemma 15's proof uses; taken literally, negative latent-factor components would be thresholded to zero and support recovery would fail, which matters for the FRED-MD application where negative coefficients appear in Figure 2. Since the proof clearly uses |bmu_ij|, this is likely a display error, but it should be corrected. Second, Theorem 3 states an error term t_k = O_p(...) without explicitly requiring that bound to be o_p(1); the asymptotic normality conclusion needs a stated condition such as m^{1/2} d*_1 d*_r^{-2} kappa_n plus the remaining terms being o(1). This is a completeness issue, not a sign that the argument is wrong. Overall, the central strong-orthogonality claim is plausible and the weak-orthogonality claim is conditional on the companion paper's Lemma 10, which is exactly the reader's CONDITIONAL verdict, so no adjustment is needed.","tokens_in":76856,"tokens_out":11135,"duration_ms":112246,"concrete_test":"Verify Lemma 10 by obtaining the proof of Theorem 3 in the companion preprint [24] and checking line-by-line that it implies sqrt(n) a^T(bmu_k - mu*_k) = h_uk + t_uk with h_uk exactly as quoted, under Conditions 1-3 and 5 of the present paper, including the sparsity requirement on a and the rate at which t_uk = o_p(1). Then substitute this expansion into the decomposition of A11 in the proof of Lemma 9 (Section C.9) and recompute the remainder t'_k; if the substitution changes the displayed distribution term or leaves a term that is not o_p(1) under Conditions 5-6, Theorem 3's asymptotic normality is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The dominant claim for the weakly orthogonal case, Theorem 3, is established by decomposing sqrt(n) a^T(bv_k - v*_k) in the supplement and controlling five error terms via Lemmas 5-9. The decisive term is Lemma 9, which converts the contribution of the hard-thresholded left-factor estimates into normal variates h_ui((Sigma_hat mu*_k)_{t_i}) plus a remainder. That conversion invokes Lemma 10, stated verbatim as 'Theorem 3, Zheng et al. [24]' and not proved in the present manuscript. Lemma 10 supplies both the asymptotic normality and the exact covariance structure of the debiased left-factor estimators bmu_i; all of Section B's variance formula (A.69) and Theorem 4's consistency proof inherit those quantities. If Lemma 10 is false, or if its conditions are not satisfied by the present paper's setup, then the h_ui terms in Theorem 3 are not the right distribution terms, the covariance matrix in (A.69) is not the limiting covariance, and Theorem 4's variance estimator is not justified. The same dependency propagates to the support-recovery Lemma 15, which uses the root-n normality of bmu_kj from Lemma 10. This is not a criticism of the companion paper's methods; it is a statement that the weak-orthogonality branch of the present paper is conditional on an unverified imported result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops SOFARI-R, a debiased-inference framework for the latent right factor vectors v*_k in the sparse SVD of a high-dimensional regression coefficient matrix C* = sum_i d*_i l*_i r*_i^T. Two variants are introduced: SOFARI-Rs for strongly orthogonal factors, which couples left singular vectors with the design matrix and rescales them to build Stiefel-manifold score functions, and SOFARI-R for weakly orthogonal factors, which removes other layers using hard-thresholded SOFARI debiased left-factor estimates. Theorems 1-2 establish asymptotic normality and variance estimation for the strong case; Theorems 3-4 do the same for the weak case. The paper reports simulation studies with coverage close to nominal and a FRED-MD forecasting application.","tokens_in":77104,"tokens_out":10200,"duration_ms":99006,"significance":"The paper addresses a genuinely asymmetric inference problem: right singular vectors cannot be handled by transposing the SOFARI construction, because layer signals are correlated through the design matrix. The coupling-and-rescaling idea for strong orthogonality is novel and avoids inverse Hessian or precision-matrix estimation in that variant. If the weak-orthogonality theorem is valid, the method provides distributional inference for all significant right factors in this model class. The supplement gives detailed proofs and lemmas for the strong case and for most of the weak-case remainder analysis. However, the weak-case theory depends on an unproved imported lemma from a companion preprint, and Theorem 3 lacks explicit rate conditions that make its stated error term o_p(1); these issues need resolution before the central claims can be accepted.","major_comments":[{"comment":"Theorem 3 states that t_k = O_p( m^{1/2} d*_1 d*^{-2}_{r*} kappa_n + s_u^{1/2} r* (d*_1 d*^{-2}_{r*} gamma_n + kappa'_n max{1, d*_{r*}^{-1}, d*_{r*}^{-2}}) ), but it does not state conditions under which this bound is o_p(1). Asymptotic normality requires t_k = o_p(1); without such conditions the displayed result is only a rate bound for an error term of unknown size. In addition, the proof of Lemma 9 in Supplement C.9 applies Lemma 10 to the sparse vector (Sigma_hat mu*_k)_{t_i}, whose l0-norm is at most s_u. That application requires s_u^{1/2} kappa_n^{(i)} = o(1), a condition that is not stated in Theorem 3 and is not shown to follow from Conditions 1-3, 5-6. Please add explicit rate conditions and verify that they hold in the simulation settings.","section":"Section 3.3 and Supplement A.3 (Theorem 3)"},{"comment":"The weak-orthogonality branch is built on Lemma 10, quoted verbatim as Theorem 3 of the authors' companion preprint [24]. This imported lemma supplies the asymptotic normality and covariance of the debiased left-factor estimators, which enter the distribution term h_{ui} in Theorem 3, the variance formula (A.69), the support-recovery Lemma 15, and the error bound in Theorem 4. The present manuscript neither proves Lemma 10 nor verifies that the companion paper's conditions match this setup. Since [24] is an arXiv preprint and not a peer-reviewed reference at the time of submission, the weak-orthogonality results are conditional on an unverified imported result. The authors should either provide a self-contained proof of Lemma 10 in the supplement or rely on a published version with explicitly checked conditions.","section":"Supplement B, Lemma 10 (and its use in Lemma 9)"},{"comment":"Lemma 15's support recovery and the hard-thresholding step rely on Condition 6, which requires signal strengths at least C_u n^{-alpha} with alpha < 1/2 and negligible total off-support mass. The paper does not discuss whether this condition is plausible for the FRED-MD design, nor whether the threshold log n / sqrt(n) is appropriate for n = 654. The confidence sets for the weakly orthogonal variant in Section 5 therefore depend on an unexamined signal-separation assumption. At minimum, the authors should report a diagnostic or state explicitly how many components are above the threshold and discuss the plausibility of Condition 6 for the economic data.","section":"Condition 6 and Section 5 (FRED-MD application)"}],"minor_comments":[{"comment":"The notation d*_{k*} appears in the statement of Lemma 8 and in its proof, but d*_{k*} is not defined. It should be d*_{r*} (or d*_k) consistently.","section":"Supplement C.8, Lemma 8"},{"comment":"In the variance formula of Lemma 10, the symbols z*_{kk} and Sigma_hat*_k are undefined. They should be mu*_k^T Sigma_hat mu*_k and Sigma_hat mu*_k r*_k^T, respectively, matching the corresponding expression in Lemma 9.","section":"Supplement B, Lemma 10"},{"comment":"The introduction and Table 1 state that SOFARI-R does not require an approximate inverse of Sigma_hat, but Definition 2 and Section 2.2 show that the weakly orthogonal SOFARI-R does use Sigma_hat via the debiased left-factor estimators. Please clarify that this advantage holds only for the strongly orthogonal SOFARI-Rs variant.","section":"Section 1 and Table 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a statistical methodology journal, and the strong-orthogonality part is well developed. I would not recommend acceptance before the weak-orthogonality branch is made self-contained: either Lemma 10 must be proved in the supplement, or [24] must be formally published with matching assumptions. The missing rate conditions in Theorem 3 are also essential for the stated asymptotic normality."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things before you read it. First, this is a serious, careful extension of the SOFARI framework, with real new machinery for inference on latent right factor vectors in sparse SVD regression. Second, the weakly orthogonal version of the main theorem borrows its central distributional result from the authors' own companion preprint, and that lemma is neither proved nor checked here.\n\nWhat is genuinely new: the coupling-and-rescaling construction (using X l*_i normalized as the factor rather than l*_i) is clever and effective. It creates a Stiefel manifold on which Neyman near-orthogonality holds, and it removes the need for approximate inverse precision matrices in the strong-orthogonality case. The strong-orthogonality theorem gives asymptotic normality for all significant right factor vectors, not just the first eigenvector, which is a real step beyond debiased sparse PCA. The supplement is detailed: each remainder term gets a separate bound, and the main bias decomposition looks internally consistent.\n\nThe load-bearing issue is Theorem 3. Lemma 9, which converts the hard-thresholded left-factor contribution into normal variates, invokes Lemma 10, stated verbatim as Theorem 3 of Zheng et al. [24] and not proved in this manuscript. That lemma supplies both the asymptotic normality and the exact covariance structure of the debiased left-factor estimators, and all of the weakly orthogonal variance formulas in (A.69) inherit those quantities. If Lemma 10 fails, or if its conditions are not satisfied in this paper's setup, Theorem 3's expansion collapses. This is not a minor gap; it is a load-bearing dependency.\n\nSmaller issues: the main text defines the hard-thresholded estimator with bmu_ij 1(bmu_ij >= log n / sqrt n), dropping the absolute value used in the supplement. Minor but confusing. The simulations are clean and coverage is good, but they stay at n=200, p,q in the 25-50 range; that does not really stress the p,q >> n regime the theory claims. There are no baseline comparisons and no released code.\n\nWho is this for? People working on high-dimensional multi-response regression and debiased sparse PCA. If the companion SOFARI paper is solid, this completes the picture and extends inference to the whole SVD. It deserves a serious referee: the strong-orthogonality contribution stands on its own, and the weak-orthogonality branch is checkable if the authors supply the missing lemma proof. I would send it out, conditional on the authors proving or properly importing Lemma 10, fixing the thresholding display, and ideally adding a higher-dimensional simulation and code release.","headline":"Solid extension of SOFARI to right-factor inference, but the weakly orthogonal branch leans on an unproved imported lemma from the companion paper.","tokens_in":77701,"tokens_out":2109,"would_cite":true,"duration_ms":22278,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H25","62F12","62J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Bias-corrected estimators give valid confidence intervals for latent right factor vectors in sparse SVD multi-response regression.","keywords":["multi-task learning","sparse singular value decomposition","manifold-based inference","Stiefel manifold","Neyman orthogonality","bias correction","asymptotic normality","high-dimensional regression"],"falsifier":"Run the weakly orthogonal SOFARI-R on a design where the cross-layer correlations $|l_j^{\\top}\\hat{\\Sigma}l_k|$ decay exactly like $c n^{-1/2}$ (so weak orthogonality fails at the boundary) and where singular-value gaps are only as large as Condition 3 requires; if the coverage of nominal 95% intervals over many replications falls well below 0.93, the $o_p(1)$ remainder in Theorem 3 is not what the proof claims. A second check is to test Condition 6 directly in the paper's own simulations: if $\\min_{j\\in\\operatorname{supp}(\\mu_i^*)}|\\mu_{ij}^*|\\ge C n^{-\\alpha}$ with $\\alpha<1/2$ does not hold, the hard-thresholding lemma cannot justify the support recovery.","tokens_in":76577,"feed_emoji":"📊","tokens_out":13396,"duration_ms":108871,"temperature":0.7,"pith_summary":"This paper tries to establish valid statistical inference—confidence intervals and tests—for the latent right factor vectors in the sparse singular value decomposition of a high-dimensional multi-response regression coefficient matrix. The difficulty is that, from the response side, signals from different SVD layers are correlated through the design matrix, so the usual manifold constraint on left singular vectors does not separate the nuisance layers. The authors couple each left singular vector with the design matrix, rescale the coupled object to unit length, and constrain it to a Stiefel manifold; a Neyman near-orthogonal score then produces bias-corrected estimators. The paper proves asymptotic normality for a strong-orthogonality variant and, using hard-thresholded debiased left-factor estimates with layer subtraction, for a weak-orthogonality variant, both with consistent variance estimates. If the theorems hold, any sparse linear combination of each latent right factor vector has a valid asymptotic normal pivot, making response selection with uncertainty quantification possible in multi-task learning and including high-dimensional PCA as a special case.","feed_headline":"Latent response factors get valid confidence intervals","feed_subtitle":"Two debiased variants handle strong and weak factor orthogonality, covering every layer of the sparse SVD.","key_machinery":"The load-bearing construction is the rescaled coupled left factor $u_i=(l_i^{\\top}\\hat{\\Sigma}l_i)^{-1/2}n^{-1/2}Xl_i$, paired with $v_i=(l_i^{\\top}\\hat{\\Sigma}l_i)^{1/2}d_i r_i$; requiring $u_i^{\\top}u_i=1$ puts these $u_i$ on Stiefel manifolds—sets of unit-length vectors—even though the original left singular vectors are not orthogonal after the design map. The constrained least-squares loss $L(v_k,\\eta_k)=(2n)^{-1}\\|Y-\\sum_i\\sqrt{n}\\,u_i v_i^{\\top}\\|_F^2$ with $u_i^{\\top}u_i=1$ and $v_i^{\\top}v_j=0$ generates a modified score $\\tilde{\\psi}_k=\\partial L/\\partial v_k-M^{(k)}\\partial L/\\partial\\eta_k$; Propositions 6 and 7 give the matrices $M^{(k)}$ and $W_k$ so that the score is Neyman near-orthogonal on the manifold and the bias term in the debiased estimate vanishes. For weakly orthogonal factors, the loss subtracts the other layers using hard-thresholded debiased left-factor estimates $\\hat{\\mu}_i^t$, and Propositions 3 and 4 give $M_k=-(\\hat{v}_k^{\\top}\\hat{v}_k)^{-1}\\hat{v}_k\\hat{u}_k^{\\top}$ and a closed-form $W_k$; the approximation errors from this substitution are absorbed into the asymptotic expansion rather than ignored.","core_discovery":"On its own terms, the paper establishes that for each layer $k$, the debiased estimator $\\hat{v}_k$ satisfies $\\sqrt{n}\\,a^{\\top}(\\hat{v}_k-v_k^*)=h_k+t_k$ for any direction $a$ with $\\|a\\|_0\\le m$, where the leading term $h_k$ is asymptotically $N(0,\\nu_k^2)$ with explicit variance $\\nu_k^2$ and the remainder satisfies $t_k=o_p(1)$ under the stated conditions. The strong-orthogonality variant attains this with a simpler variance and no precision-matrix estimation; the weak-orthogonality variant attains it with a variance that also includes the debiased left-factor estimates of the other layers. Theorems 2 and 4 provide plug-in variance estimates whose error is of the same order as the SOFAR estimation error, so the pivots can be used to build confidence intervals for every component of every significant right factor vector. This completes inference on all SVD components of the coefficient matrix, after earlier work covered the left factor vectors and singular values.","pith_inferences":["The per-layer expansions suggest joint confidence regions across several right factor vectors could be built from the same normal terms, but the paper does not develop simultaneous inference.","Because the strong-orthogonality variant avoids precision-matrix estimation, the coupling-rescaling construction may transfer to designs where the Gram matrix is singular or the precision matrix is not sparse.","The hard-thresholding step offers a diagnostic: if the support of the thresholded left-factor estimate changes when the threshold is moved around $\\log n/\\sqrt{n}$, Condition 6 may be violated and weak-orthogonality intervals deserve caution.","Theorem 3 inherits any unproved gap in the companion lemma on debiased left-factor normality; a self-contained proof of that lemma inside this framework would remove the main external dependency."],"forward_implications":["For any layer and any sparse direction, the debiased estimator provides an asymptotically normal pivot for the corresponding linear combination of the latent right factor vector, so componentwise confidence intervals reach the nominal coverage as the sample size grows.","The strong-orthogonality variant needs no approximate inverse of the design Gram matrix and imposes no sparsity condition on the precision matrix, widening the settings where such inference is available.","The weak-orthogonality variant extends the method to correlated latent factors, provided the debiased left-factor distribution theory and the hard-thresholding signal-separation condition hold.","Together with inference for left factor vectors and singular values, the results complete manifold-based inference for all sparse SVD components of the coefficient matrix in multi-response regression.","When the design matrix is the identity, the right factor vectors are principal components, so the strong-orthogonality variant gives asymptotic distributions for all significant principal-component directions, not just the first eigenvector."],"supporting_citations":[{"why":"Supplies the companion SOFARI debiased left-factor estimators whose asymptotic normality is imported as Lemma 10 in the weakly orthogonal variant.","marker":"[24]"},{"why":"Provides the SOFAR estimator and its sparse-SVD error bounds used as the initial estimate throughout both variants.","marker":"[17]"},{"why":"Gives the Neyman near-orthogonality framework that motivates the modified score construction.","marker":"[9]"},{"why":"Debiased sparse PCA baseline; its first-eigenvector inference is extended here to all significant eigenvectors when the design is the identity.","marker":"[12]"},{"why":"Adaptive thresholding method used to estimate the error covariance matrix for the plug-in variance estimators.","marker":"[4]"},{"why":"Nodewise Lasso used to estimate the precision matrix inside the debiased left-factor step for weakly orthogonal factors.","marker":"[14]"},{"why":"Rank-selection method used to estimate the number of SVD layers before fitting the SOFAR initial estimate.","marker":"[2]"},{"why":"FRED-MD macroeconomic data set used in the application.","marker":"[13]"}],"fun_headline_variants":["Right factor vectors now have valid CIs","Two paths to debiased right factor inference","Completing SVD inference: right factors covered","Strong or weak, right factors get valid CIs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the companion paper's debiased left-factor estimators $\\hat{\\mu}_i$ are asymptotically normal under the stated conditions (imported as Lemma 10), and that the hard-thresholding signal-separation Condition 6 holds, since Theorem 3's expansion collapses if either fails.","fun_headline_variants_meta":{"raw":{"variants":["Right factor vectors now have valid CIs","Two paths to debiased right factor inference","Completing SVD inference: right factors covered","Strong or weak, right factors get valid CIs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001062,"raw_usage":{"total_tokens":4496,"prompt_tokens":1030,"completion_tokens":3466,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":3406}},"tokens_in":646,"tokens_out":3466,"duration_ms":23294,"temperature":1.0,"reasoning_tokens":3406,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:30:37.338569+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the weakly orthogonal SOFARI-R on a design where the cross-layer correlations $|l_j^{\\top}\\hat{\\Sigma}l_k|$ decay exactly like $c n^{-1/2}$ (so weak orthogonality fails at the boundary) and where singular-value gaps are only as large as Condition 3 requires; if the coverage of nominal 95% intervals over many replications falls well below 0.93, the $o_p(1)$ remainder in Theorem 3 is not what the proof claims. A second check is to test Condition 6 directly in the paper's own simulations: if $\\min_{j\\in\\operatorname{supp}(\\mu_i^*)}|\\mu_{ij}^*|\\ge C n^{-\\alpha}$ with $\\alpha<1/2$ does not hold, the hard-thresholding lemma cannot justify the support recovery.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the SOFAR estimator and its sparse-SVD error bounds used as the initial estimate throughout both variants."},{"cited_title":"Chetverikov, M","cited_arxiv_id":null,"evidence_quote":"Gives the Neyman near-orthogonality framework that motivates the modified score construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Debiased sparse PCA baseline; its first-eigenvector inference is extended here to all significant eigenvectors when the design is the identity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Adaptive thresholding method used to estimate the error covariance matrix for the plug-in variance estimators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Nodewise Lasso used to estimate the precision matrix inside the debiased left-factor step for weakly orthogonal factors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rank-selection method used to estimate the number of SVD layers before fitting the SOFAR initial estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"FRED-MD macroeconomic data set used in the application."}],"review_version":1}