{"id":"88a7b616-43eb-4832-a2aa-7d0862fa66e7","arxiv_id":"2412.16773","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Introduces inducing-variable and frequency-domain approximations for multi-group Gaussian process factor models, achieving linear scaling in trial length and group count with minimal accuracy loss.","lead":"This paper develops two faster ways to fit a Gaussian-process factor model that tracks interactions between multiple brain areas, cutting the cost from cubic to linear in trial length and number of areas. The frequency-domain version runs about 25 times faster than the standard approach on real three-area recordings with no loss in statistical accuracy.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"mDLAG-inducing's linear-in-T claim depends on holding Tind fixed, but accuracy forces Tind to grow with T, making practical scaling superlinear; the abstract's 'both methods' linear-scaling claim overstates this method.","rationale":"The reader's weakest_assumption identified the same load-bearing premise: mDLAG-inducing's linear scaling in T assumes Tind stays fixed, while the paper's own experiments raise Tind with T to avoid aliasing, producing superlinear observed scaling. I focus on this as the single most load-bearing concern because it directly contradicts the abstract's blanket statement that both methods achieve linear scaling in trial length, and the paper's own Discussion concedes the limitation without amending the central claim. The alternate concern about frequency-domain simulation generation is mitigated in the manuscript: Methods 6.1 generates 3T-length trials via the circulant model and discards the outer thirds, which for the timescale parameters used makes the middle segment an essentially exact sample from the stationary time-domain GP; thus it is not a decisive flaw. The frequency-domain method's real-data results (Fig. 5) and its internal bias characterization provide independent support, so the overall verdict should remain conditional rather than be rejected. The proposed concrete test would settle whether the inducing-variable method's linear-scaling claim can be salvaged in any accuracy-preserving regime, or whether the abstract must be qualified to attribute linear scaling only to mDLAG-frequency.","tokens_in":56597,"tokens_out":9247,"duration_ms":87584,"concrete_test":"Repeat the Section 4.2 trial-length scaling experiment (M=2, p=1, squared-exponential GP with 100 ms timescale, T from 50 to 500) and, for each T, measure the minimal Tind required to keep leave-unit-out R2 within 1% of the mDLAG-time baseline. If this minimal Tind grows roughly linearly with T, then the practical per-iteration cost of mDLAG-inducing is O(T^3), and the paper's linear-in-T claim for this method must be retracted or explicitly restricted to fixed-Tind regimes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that both accelerated methods scale linearly in trial length is insecure for mDLAG-inducing. In Section 3.2.2 and Appendix B, the per-iteration cost is O(p^3 M T Tind^2) plus O(p^3 Tind^3), which is linear in T only for fixed Tind. However, Section 4.2 states that Tind was increased with T to keep the induced sampling rate above the approximate Nyquist rate, and without this, mDLAG-inducing performance degrades (Supplementary Fig. 2). Consequently, Fig. 3B shows mDLAG-inducing scaling superlinearly with T. If Tind must be proportional to T to avoid aliasing, the per-iteration cost becomes O(p^3 M T^3), erasing the advertised linear-in-T advantage. On real data, the trade-off is explicit: Tind=20 gives a speed similar to mDLAG-frequency but significantly worse leave-group-out R2 (Fig. 5B), while Tind=32 restores statistical performance but reduces the speed advantage (Supplementary Fig. 4). Thus, for mDLAG-inducing there is no demonstrated operating point where linear-in-T scaling and accuracy comparable to the time-domain baseline hold simultaneously. The frequency-domain method remains supported, but the abstract's claim that 'both methods' improve scaling from cubic to linear in trial length is not justified for mDLAG-inducing under the conditions needed for accurate inference.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two approximate inference algorithms for multi-group Gaussian process factor models of the type used in multi-area neural recordings: mDLAG-inducing, which introduces a shared inducing variable per latent, and mDLAG-frequency, which formulates the model directly in the Fourier domain under a diagonal-spectrum (circulant) approximation to the GP covariance. The authors derive variational updates and gradient-based GP parameter updates for both methods, state per-iteration complexity claims of linear scaling in trial length T and group number M, and evaluate the methods on simulated data and Neuropixels recordings from three visual cortical areas. They also characterize finite-T biases in mDLAG-frequency's GP timescale, time delay, and dimensionality estimates and show that tapering mitigates these biases. The central claims are that both accelerated methods preserve the statistical performance of the time-domain baseline while achieving orders-of-magnitude speed-ups, with mDLAG-frequency offering the best overall trade-off.","tokens_in":56986,"tokens_out":7706,"duration_ms":72935,"significance":"If the speed and accuracy claims hold as stated, the paper would make a practically important class of dimensionality-reduction methods feasible for large multi-population recordings. The variational derivations in Appendices B and C are detailed, the scaling analyses are internally consistent under their stated assumptions, and the frequency-domain method is supported by a thorough bias analysis and a reproducible-looking evaluation protocol. However, the paper's abstract and introduction overstate the linear-scaling evidence for mDLAG-inducing, and the frequency-domain method's finite-T parameter biases are substantial enough that the blanket claim of 'minimal impact on statistical performance' needs sharp scoping. With those qualifications addressed, the contribution is significant for the computational neuroscience and GP latent-variable-model communities.","major_comments":[{"comment":"The Discussion repeats the overstatement: 'We demonstrated that mDLAG-inducing and mDLAG-frequency can achieve linear scaling in both the number of time points per trial T and the number of groups M'. This is true for mDLAG-inducing only conditionally on Tind being fixed, and the subsequent paragraphs correctly describe the practical trade-off. The summary paragraph should be amended so that the main claim matches the evidence presented in Sections 4.2 and 4.4.","section":"Section 5, first paragraph"}],"minor_comments":[{"comment":"The text says mDLAG-frequency 'performed statistically as well as mDLAG-time' based on Fig. 5B, but no paired statistical test is reported for that comparison; adding the sign-test p-value or a confidence interval would make the claim precise.","section":"Section 4.4"},{"comment":"The latent time-course R2 values are computed after re-inferring all latents with mDLAG-time inference (equation 3.13), even for models fit by mDLAG-inducing and mDLAG-frequency. This is a reasonable way to compare parameter estimates, but it does not evaluate the latent estimates users would obtain from each method's native inference. A sentence stating this explicitly would prevent misinterpretation.","section":"Section 4.1; Fig. 2C"},{"comment":"The implementations of mDLAG-inducing and mDLAG-frequency are listed as 'will be made available upon publication'; making the code available at submission would strengthen the reproducibility of the empirical claims.","section":"Reproducibility section"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong fit for a computational statistics or statistical machine learning venue. The primary issue is an overbroad central claim about mDLAG-inducing's linear scaling, which the authors themselves partially qualify in the Discussion; this is fixable by rewording the abstract and introduction. The frequency-domain method is the more defensible contribution and is well supported by the bias analysis. I recommend major revision rather than rejection, as I see no load-bearing mathematical error that would require starting over."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The frequency-domain method is the real contribution. It takes the existing Whittle-likelihood idea and assembles it properly for multi-group GP factor models with per-group delays, getting O(MT) scaling, and the Neuropixels validation is convincing: 25x overall speed-up with leave-group-out R2 statistically indistinguishable from the time-domain baseline. The bias analysis (circulant approximation, timescale/delay underestimation, dimensionality inflation at short T) and the tapering mitigation are genuinely useful.\n\nThe inducing-variable method is weaker than the abstract suggests. Linear-in-T only holds for fixed Tind, but accuracy forces Tind to grow with T to avoid aliasing; the paper's own Fig. 3B shows superlinear scaling, and on real data Tind=20 gives speed but significantly worse performance while Tind=32 restores performance but erodes the speed advantage. The paper discloses this in the Discussion, but the abstract says both methods achieve linear scaling and 'minimal impact on statistical performance,' which is misleading for mDLAG-inducing.\n\nMinor concern: large-scale simulations are generated from the frequency-domain model, so they can't independently validate that method's approximation error at scale. The real data partially covers this, and the edge-effect mitigation is careful, so I'd call it minor.\n\nOverall, this is a solid methods paper. The frequency-domain approach deserves to be in the toolbox, and the bias characterization is more careful than most. The main fix before publication is to qualify the abstract and clarify that the linear-scaling claim for mDLAG-inducing requires Tind to be fixed, with accuracy-dependent caveats. I'd send it to serious peer review and expect it to be accepted with moderate revision. I'd cite the frequency-domain part in my own work.","headline":"Frequency-domain acceleration is solid and useful; the abstract overstates the inducing-variable variant.","tokens_in":57447,"tokens_out":2330,"would_cite":true,"duration_ms":21473,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G15","62F15","62H25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that multi-group Gaussian process factor models can be fit in time linear in trial length and group count using inducing variables or a frequency-domain approximation, with orders-of-magnitude speedups and negligible…","keywords":["Gaussian process factor models","multi-group dimensionality reduction","inducing variables","frequency domain","Whittle likelihood","linear-time scaling","neural population recordings","automatic relevance determination"],"falsifier":"In the paper's own trial-length sweep, mDLAG-frequency's leave-group-out R2 is already about 1 percent below mDLAG-time at T=10 (Fig. 3A inset); a dataset with short trials and fast latent timescales where this gap reaches a practically meaningful level at T at least 100, or where estimated GP timescales stay biased by more than about 10 percent even at long T, would falsify the claim that the frequency-domain fit preserves statistical performance.","tokens_in":56410,"feed_emoji":"⚡","tokens_out":6736,"duration_ms":50854,"temperature":0.7,"pith_summary":"The paper sets out to remove the computational bottleneck in a class of dimensionality-reduction models used to study how multiple neural populations interact over time. The baseline method, called mDLAG-time, must invert covariance matrices that grow cubically in the number of time points per trial and in the number of recorded groups, which makes large multi-area recordings impractical. The paper proposes two approximations: one that passes latent structure through a small set of inducing variables, and one that formulates the entire generative model in the frequency domain, where the GP covariance becomes diagonal. Both achieve linear scaling in trial length and group number. On simulated data and on Neuropixels recordings of hundreds of neurons across three visual areas, the frequency-domain method delivers the largest speedups (up to about 25x overall on real data) while matching the predictive performance of the exact time-domain fit.","feed_headline":"Frequency-domain fitting speeds up multi-group GP factor models 25x","feed_subtitle":"Two approximations cut cubic scaling to linear and match time-domain accuracy on three-area neural recordings.","key_machinery":"The load-bearing object is the frequency-domain (Whittle) approximation to the Gaussian process state model: for a stationary GP, the unitary DFT of the covariance matrix converges to a diagonal matrix whose entries are the power spectral density, so the multi-group covariance is replaced by a diagonal PSD matrix. Because the time-delay terms factor as a product of per-group phase-shift matrices, the posterior covariance at each frequency is a p by p matrix inverse rather than a pMT by pMT inverse, which is what turns O($M^{3}$ $T^{3}$) into O(M T) per iteration. The inducing-variable variant instead replaces the MT-dimensional latent trajectory with a small number of inducing points per latent, giving O(M T $T_ind^{2}$) operations and preserving the ARD-based group-sparsity structure of the original model.","core_discovery":"The central claim is that the cubic runtime scaling that has limited multi-group Gaussian process factor models can be broken: mDLAG-inducing, built on variational inducing variables, and mDLAG-frequency, a fully frequency-domain generative model, both reduce the cost of the three bottleneck stages of fitting (latent posterior update, GP-parameter gradients, and ELBO evaluation) to O($p^{3}$ T + $p^{2}$ M T) and O(M T) per iteration respectively, i.e., linear in the number of time points per trial T and the number of groups M. The frequency-domain construction is the more complete of the two: by taking the GP covariance to be diagonal in the Fourier basis (the Whittle / circulant approximation), the posterior over latents factorizes across frequencies, and the time-delay parameters enter only as diagonal phase-shift matrices, so each frequency update costs O($p^{3}$) plus O($p^{2}$ M). Empirically the paper shows that mDLAG-frequency matches mDLAG-time's leave-group-out R2 on long simulated trials and on three-area Neuropixels data while running about 25x faster end-to-end, and it characterizes the cost of the approximation: at short trial lengths, GP timescales and delays are underestimated and latent dimensionality can be overestimated, biases that shrink as T grows and that tapering largely removes.","pith_inferences":["The frequency-domain method's scaling advantage grows exactly where its approximation is most accurate, so the practical sweet spot is long trials; a hybrid protocol of fitting with mDLAG-frequency for fast exploration and then fine-tuning the retained model with mDLAG-time could capture the best of both.","Because the same diagonal-covariance trick is what makes the approach linear in the number of groups, the method should extend to non-Gaussian observation models (e.g., Poisson spike counts) via a frequency-domain quasi-likelihood, though the paper only demonstrates the Gaussian case.","The bias characterization implies that reported GP timescales and delays from frequency-domain fits should be interpreted with a finite-T correction, and that cross-validated dimensionality selection via ARD may overstate the number of shared latents on short trials.","The inducing-variable approach's need to keep the number of inducing points above the Nyquist rate couples its runtime to the fastest latent timescale, so on data with fast timescales the frequency-domain method is likely to remain the better default, a hypothesis testable by comparing the two on a dataset with known fast and slow latents."],"forward_implications":["Multi-group GP factor models can be applied to trials with hundreds of time points and to recordings spanning many brain areas, where the baseline method would require days of compute.","The frequency-domain fit can replace the time-domain fit in exploratory analyses: on the datasets tested, leave-group-out R2 is statistically indistinguishable while runtime drops by orders of magnitude.","The same two approximation strategies carry over to the broader GP factor model family (GPFA, DLAG, group factor analysis), since mDLAG subsumes them as special cases.","Users of these methods must be aware of the frequency-domain biases at short trial lengths, and can mitigate them with tapering or by fine-tuning with the time-domain model."],"supporting_citations":[{"why":"Defines the baseline mDLAG model whose cubic scaling this paper targets.","marker":"(Gokcen et al., 2023)"},{"why":"Supplies the variational inducing-variable framework mDLAG-inducing is built on.","marker":"(Titsias, 2009)"},{"why":"Extends inducing-variable approximations to multi-output GPs, giving the O(M T T_ind^2) complexity.","marker":"(´Alvarez et al., 2010)"},{"why":"Establishes the frequency-domain quasi-likelihood that justifies the diagonal PSD approximation.","marker":"(Whittle, 1951)"},{"why":"Provides the spectral theory for diagonalization of stationary GP covariances and the tapering bias-mitigation strategy.","marker":"(Priestley, 1981)"},{"why":"Introduces frequency-domain quasi-likelihood fitting for GP factor models, the precursor of mDLAG-frequency.","marker":"(Keeley, Aoi, et al., 2020)"},{"why":"Formulates multi-output GP kernels through phase shifts, the basis for the time-delay terms in the frequency domain.","marker":"(Ulrich et al., 2015)"},{"why":"Supplies the group factor analysis with ARD whose variational updates and sparsity priors mDLAG adapts.","marker":"(Klami et al., 2015)"}],"fun_headline_variants":["Frequency-domain method makes multi-group GP factor models 25x faster","Linear scaling for multi-group GP factor models via frequency domain","25x speedup: frequency-domain approach for multi-group GP factor models","Multi-group GP factor models: from cubic to linear in frequency domain"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central speedups rest on two approximations whose error grows as trials get shorter: the frequency-domain method assumes the GP covariance is diagonal (circulant in time, exact only as T grows), and the inducing-variable method assumes the number of inducing points can stay fixed without aliasing fast latent timescales.","fun_headline_variants_meta":{"raw":{"variants":["Frequency-domain method makes multi-group GP factor models 25x faster","Linear scaling for multi-group GP factor models via frequency domain","25x speedup: frequency-domain approach for multi-group GP factor models","Multi-group GP factor models: from cubic to linear in frequency domain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1776,"prompt_tokens":1035,"completion_tokens":741,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":667}},"tokens_in":651,"tokens_out":741,"duration_ms":6256,"temperature":1.0,"reasoning_tokens":667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:15:31.295967+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the paper's own trial-length sweep, mDLAG-frequency's leave-group-out R2 is already about 1 percent below mDLAG-time at T=10 (Fig. 3A inset); a dataset with short trials and fast latent timescales where this gap reaches a practically meaningful level at T at least 100, or where estimated GP timescales stay biased by more than about 10 percent even at long T, would falsify the claim that the frequency-domain fit preserves statistical performance.","supporting_citations":[{"cited_title":"APACrefauthors \\ 2009","cited_arxiv_id":null,"evidence_quote":"Supplies the variational inducing-variable framework mDLAG-inducing is built on."},{"cited_title":", Luengo, D","cited_arxiv_id":null,"evidence_quote":"Extends inducing-variable approximations to multi-output GPs, giving the O(M T T_ind^2) complexity."},{"cited_title":"APACrefauthors \\ 1951","cited_arxiv_id":null,"evidence_quote":"Establishes the frequency-domain quasi-likelihood that justifies the diagonal PSD approximation."},{"cited_title":"APACrefauthors \\ 1981","cited_arxiv_id":null,"evidence_quote":"Provides the spectral theory for diagonalization of stationary GP covariances and the tapering bias-mitigation strategy."},{"cited_title":", Carlson, D E","cited_arxiv_id":null,"evidence_quote":"Formulates multi-output GP kernels through phase shifts, the basis for the time-delay terms in the frequency domain."}],"review_version":1}