{"id":"95cc36f2-20a8-4da1-9a7e-3e1bea5b5605","arxiv_id":"2506.05065","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"UnHiPPO uses a Kalman filter view of HiPPO to derive a noise-robust initialization for state space models that filters measurement noise with no added runtime.","lead":"This paper introduces a new way to initialize state space models by treating noisy input data as noisy measurements of a hidden system, then using a Kalman filter to clean the signal. The authors show that this initialization improves how well the models handle noisy audio and time series data without slowing them down.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Eq. (11) substitution makes UnHiPPO's 'posterior inference' claim conditional on an approximation that is only exact for polynomial signals; a synthetic oracle check is needed to settle whether the method actually denoises the true HiPPO coefficients.","rationale":"The reader's weakest_assumption pinpoints the same substitution in Eq. (11) and the ad hoc pseudo-inverse regularization, and that is the most load-bearing threat to the paper's theoretical framing. If the substitution is a poor approximation for noisy non-polynomial signals, then UnHiPPO is not performing posterior inference over the HiPPO projection coefficients, even though the Kalman update is algebraically correct for the regularized model. That would weaken the central claim's motivation without necessarily invalidating the empirical improvement. The proposed synthetic test settles this directly and is cheap to run, since the ground-truth projection coefficients are computable. The empirical results are suggestive but not independently verified with error bars in the main comparisons, so keeping the reader's conditional verdict is appropriate: the method is plausible and reproducible, but the theoretical claim needs the oracle check, and the empirical claim needs additional runs before acceptance on the stated strength.","tokens_in":16710,"tokens_out":9677,"duration_ms":120737,"concrete_test":"On synthetic signals drawn from a Gaussian process with known covariance, add Gaussian noise at the levels used in Fig. 6. Compute the exact running Legendre projection coefficients c_t^* of the clean signal by numerical quadrature. Run (a) the standard HiPPO recurrence, (b) the time-varying UnHiPPO Kalman recurrence from Eq. (24), and (c) the time-invariant UnLSSL initialization on the noisy observations. Report the mean squared error of each state sequence relative to c_t^*. If (b) and (c) do not substantially reduce this error relative to (a) at the noise levels tested, the posterior-inference interpretation fails and the observed accuracy gains must be attributed to something other than denoising the true projection coefficients.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central derivation relies on replacing f(t) in the HiPPO ODE (Eq. 4) with the reconstruction B_H^T c_t to obtain the data-free dynamics in Eq. (11). This substitution is exact only when the signal equals its own N-term Legendre projection; for general noisy signals it defines a different latent process, not the evolution of the true projection coefficients. The pseudo-inverse regularization in Eq. (18) is then an additional heuristic modification of that already approximate process. As a result, the Kalman filter in Eqs. (22)-(23) computes the posterior of a state whose dynamics were not derived from a correct generative model of the HiPPO coefficients. The paper's claim that UnHiPPO 'infers the posterior of this latent system' and thereby extends HiPPO theory to noisy data is therefore not actually established. The empirical improvements in Figs. 6-7 may still hold, but the theoretical justification attached to the central claim is not secure without checking how far the state produced by UnHiPPO is from the true projection coefficients of the clean signal.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes UnHiPPO, an alternative initialization for state space models (SSMs) that is meant to be robust to measurement noise. The authors reinterpret the HiPPO dynamics as a linear stochastic control problem, replace the noise-free control input by a noisy observation model, regularize the resulting data-free dynamics via extrapolation constraints and a pseudo-inverse, and then use a Kalman filter to derive a closed-form linear recurrence for the posterior mean. The resulting UnHiPPO matrices are used to initialize an LSSL architecture (UnLSSL). Experiments on two speech classification datasets compare UnLSSL against LSSL under various training and evaluation noise levels.","tokens_in":17041,"tokens_out":5332,"duration_ms":68442,"significance":"If the central claim holds, UnHiPPO is an attractive drop-in replacement for the standard HiPPO initialization: it does not change the SSM structure, does not increase runtime, and provides a tunable noise-filtering behavior through a single scalar hyperparameter. The paper is transparent about some limitations, most notably the absence of a structured parametrization for large state dimensions, and it provides a public implementation link. However, the theoretical claim that UnHiPPO performs posterior inference for the HiPPO coefficients is not fully established, because the derivation replaces the observed signal by its polynomial reconstruction at a load-bearing step. The empirical results are promising but lack error bars, so the strength of the reported improvements is difficult to assess.","major_comments":[{"comment":"The substitution f(t) = B_H^T c_t in Eq. (11) is exact only when the observed signal equals its N-term Legendre projection. For general noisy signals, the resulting data-free dynamics define a different latent process, not the evolution of the true HiPPO projection coefficients. Consequently, the Kalman filter in Eqs. (22)-(23) computes the posterior of the regularized surrogate system, and the paper's claim that UnHiPPO extends HiPPO theory to noisy data by performing posterior inference is not established as stated. I recommend adding a synthetic oracle experiment in which the ground-truth projection coefficients of a clean signal are known, comparing the UnHiPPO state against those coefficients under various noise levels, or explicitly reframing the method as a heuristic filter motivated by, rather than derived from, HiPPO.","section":"Section 4, Eq. (11)"},{"comment":"The pseudo-inverse regularization in Eq. (18) is introduced as an ad hoc modification: the overdetermined system in Eq. (17) is solved by least squares without a clear optimality criterion tied to the noise model or to the HiPPO projection objective. This step changes the dynamics even in the noise-free polynomial case, where Eq. (11) would otherwise be exact. The paper should either provide a principled justification for the two conditions in Eq. (13) and the least-squares solution, or explicitly label the regularized dynamics as a heuristic approximation and temper the theoretical claims accordingly.","section":"Section 4.1, Eq. (18)"},{"comment":"The central empirical claim is that UnLSSL improves noise robustness, but Figs. 6 and 7 report accuracy differences without error bars and without stating the number of independent runs. At several noise levels the reported differences are small or negative, so without variance estimates it is unclear whether the improvements are statistically significant. Please report mean and standard deviation over multiple seeds, and preferably also the raw accuracies for both models, so that the magnitude of the effect can be judged.","section":"Section 6, Figs. 6 and 7"}],"minor_comments":[{"comment":"The hyperparameter table is described as applying to the SC10 experiments, but the FSD experiments do not have a corresponding table; please report the FSD hyperparameters and the number of seeds used for all experiments.","section":"Appendix C"},{"comment":"The limitation that no structured O(N) parametrization of the UnHiPPO matrices was found is important for scalability; consider mentioning this point earlier in the paper, since many modern SSM applications rely on structured representations.","section":"Section 8"},{"comment":"The index convention for the transition matrix is easy to confuse: Eq. (20) uses \\bar{A}_{R,k+1} while Eq. (22) uses \\bar{A}_{R,k} for the same transition from step k-1 to k. Please align the notation for readability.","section":"Eqs. (20) and (22)"},{"comment":"The caption of Fig. 5 is unclear about what the labels k=500, k=100, k=20 refer to and how many times the fixed dynamics are applied; please clarify the experimental protocol used for that figure.","section":"Figure 5"},{"comment":"In the provided rendering, superscripts that should denote a matrix inverse appear as the characters \"91\"; please ensure the final PDF displays the intended exponents correctly.","section":"Equations (27), (46), and (49)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a machine learning venue and the empirical direction is useful, but the theoretical framing currently overstates what is derived. The main fix is to either add the oracle validation that directly measures closeness to true HiPPO coefficients or to rewrite the contributions as proposing a heuristic noise-robust initialization motivated by Kalman filtering. With that change plus error bars, the paper would be substantially stronger."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"UnHiPPO is a genuinely new initialization for SSMs: it recasts the HiPPO update as a linear stochastic control problem, turns the signal into a noisy observation, and uses a Kalman filter to derive time-varying matrices A_U and B_U that denoise the state without changing the model or its runtime. That is a clean idea, and the paper develops it carefully. The derivation is honest: the substitution that makes the dynamics data-free in Eq. (11) is only exact for polynomial signals, and the authors say so. The pseudo-inverse regularization in Eq. (18) is a heuristic to control extrapolation, also stated. So the 'posterior inference' language in the abstract overstates what is actually established: the Kalman filter computes the posterior of an approximate model, not of the original HiPPO coefficients. That is the main soft spot, but it is not a hidden one.\n\nWhat the paper does well beyond the idea: the write-up is unusually clear, the matrix-exponential discretization is a nice practical contribution, and the experiments on two speech datasets show consistent gains when evaluation noise exceeds training noise. The limitations section is straight about not having a structured parametrization.\n\nWhere it is soft: the empirical evaluation is narrow. Only the LSSL architecture is tested, no modern SSM like S4 or Mamba, and the main comparison figures lack error bars. The sigma^2 hyperparameter needs tuning per noise level and the values are far from the actual noise variance; that weakens the 'uncertainty-aware' story. The authors should add a synthetic oracle check: generate a clean signal, add noise, run UnHiPPO, and compare the filtered state to the true projection coefficients of the clean signal. That would settle whether the method actually tracks the HiPPO latent state or just produces a smoothed version.\n\nFor a reader working on SSM initialization or noisy time series, this is worth reading. It is not a paradigm shift, but it is a solid, honest contribution. I would send it to peer review, with a request for the oracle experiment, error bars, and a comparison on at least one modern SSM. The core idea holds up.","headline":"A clean, well-motivated extension of HiPPO that trades exact theoretical grounding for a useful heuristic; worth reviewing, but the posterior-inference claim is only conditional on a polynomial-signal approximation the authors acknowledge.","tokens_in":17462,"tokens_out":2855,"would_cite":false,"duration_ms":31815,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that reinterpreting HiPPO as a linear stochastic control problem with noisy observations yields a new initialization, UnHiPPO, whose Kalman-filter recurrence makes state space models robust to measurement noise at no…","keywords":["state space models","HiPPO","Kalman filter","measurement noise","sequence classification","Legendre polynomials","model initialization","linear dynamical systems"],"falsifier":"Run the UnHiPPO recurrence on a clean polynomial signal of degree at most $N-1$, where the Legendre projection is exact, with $\\sigma^2 \\to 0$: if the posterior mean does not track the known HiPPO projection coefficients, the substitution underlying the dynamics is wrong even in the noise-free case. On synthetic signals with known ground truth (e.g., Gaussian process samples with added Gaussian noise), sweep $\\sigma^2$ and compare the reconstruction error of UnHiPPO against HiPPO: if no value of $\\sigma^2$ yields a lower error, the claimed noise filtering does not occur.","tokens_in":16551,"feed_emoji":"🎛️","tokens_out":8451,"duration_ms":84061,"temperature":0.7,"pith_summary":"HiPPO, the standard initialization for state space models, assumes the input signal is noise-free. The paper drops that assumption by rewriting HiPPO as a linear stochastic control problem and then swapping the noise-free control signal for noisy observations of a latent system. The resulting UnHiPPO initialization replaces the HiPPO matrices with a Kalman-filter update for the posterior mean of the latent polynomial coefficients. Because this is just a different initialization of the same linear recurrence, the model structure and runtime are unchanged while the model gains an explicit, tunable mechanism for filtering measurement noise.","feed_headline":"Kalman-filtered HiPPO init makes SSMs shrug off noise","feed_subtitle":"Same model and runtime: the recurrence now performs a Kalman-filter update, filtering measurement noise.","key_machinery":"The machinery is the Kalman-filter posterior-mean update reduced to a single linear recurrence (Eq. 24): $m_k = (I - K_k B_H^T)\\bar{A}_{R,k} m_{k-1} + K_k y_k$, defining the uncertainty-aware HiPPO matrices $\\bar{A}_{U,k}$ and $\\bar{B}_{U,k}$. The prior dynamics use the regularized HiPPO matrix $A_R$, obtained by solving an overdetermined system of shape-preservation conditions with a pseudo-inverse, and are discretized in closed form as the matrix exponential $\\exp(\\log(t_{k+1}/t_k) A_R)$. This turns the polynomial projection problem into a filtering problem while preserving the structure of the LSSL recurrence and its Krylov-kernel acceleration.","core_discovery":"The central claim is that the HiPPO dynamics, which treat the observed signal $f(t)$ as a noise-free control input, can be re-derived as a continuous-discrete linear dynamical system in which $f(t)$ is a noisy observation of a latent state. Substituting the Legendre reconstruction $\\hat{f}_{\\le t}(t) = B_H^T c_t$ for $f(t)$ makes the dynamics data-free, and a pseudo-inverse regularization (Eq. 18) tames the extrapolation blow-up of Legendre polynomials. Filtering this system with a Kalman filter collapses to a single linear recurrence for the posterior mean, $m_k = \\bar{A}_{U,k}m_{k-1} + \\bar{B}_{U,k} y_k$, where $\\bar{A}_{U,k}$ and $\\bar{B}_{U,k}$ define the UnHiPPO initialization. The paper argues that an SSM initialized this way implicitly performs posterior inference over the projection coefficients it would have computed under noise-free HiPPO, and that this improves classification accuracy on noisy speech data both when training and test noise are matched and when they differ, with no increase in runtime.","pith_inferences":["The control-to-inference rewrite is not specific to Legendre polynomials; the same substitution and filtering construction could be applied to other orthogonal-basis HiPPO variants, yielding uncertainty-aware versions of Laguerre- or Fourier-based memory.","Because the Kalman gain $K_k$ is data-independent, UnHiPPO is effectively a fixed linear denoiser; characterizing its frequency response (or its equivalent linear filter) could tell practitioners which noise spectra it attenuates, which the paper does not analyze.","The method is demonstrated on univariate speech waveforms with additive Gaussian noise; a natural test is whether it extends to multivariate or non-Gaussian measurement noise, where the scalar-observation and Gaussian assumptions would need to be relaxed.","The paper notes that $\\sigma^2$ needs to be very large to dominate the $B_H^T P^- B_H$ term, which suggests the noise parameter encodes numerical scale as much as true measurement variance; a reparameterization (e.g., normalizing the observation equation) could make $\\sigma^2$ interpretable as actual noise variance."],"forward_implications":["Any SSM initialized with HiPPO can switch to UnHiPPO as a drop-in replacement, gaining noise robustness with no architectural change and no runtime increase.","The $\\sigma^2$ hyperparameter gives the user a single dial that controls how much noise is filtered, letting the model adapt to the expected noise level without retraining the backbone.","The closed-form matrix-exponential discretization is numerically stable on long sequences where trapezoidal and forward Euler discretizations of $A_R$ diverge, so UnHiPPO also provides a more stable way to discretize the regularized dynamics.","Because the posterior covariance is data-independent, the entire filter is a precomputable linear operator, so the Krylov-kernel convolution acceleration used by LSSL applies unchanged to UnLSSL.","On the two speech classification benchmarks, UnLSSL improves accuracy over LSSL when evaluation noise differs from training noise, and matches or exceeds it at high training noise levels."],"supporting_citations":[{"why":"This reference supplies the HiPPO dynamics, the matrices $A_H$ and $B_H$, and the discretization that UnHiPPO modifies.","marker":"(Gu et al., 2020)"},{"why":"This reference defines the LSSL architecture and its HiPPO-based initialization scheme and Krylov-kernel acceleration that UnLSSL inherits.","marker":"(Gu et al., 2021)"},{"why":"This reference provides the continuous-discrete discretization of linear SDEs and the Kalman filter equations used to derive Eq. (24).","marker":"(Särkkä & Solin, 2019)"},{"why":"This reference introduces the Kalman filter whose posterior-mean update becomes the UnHiPPO recurrence.","marker":"(Kalman, 1960)"},{"why":"This reference frames HiPPO as a linear stochastic control problem, which motivates the noise model and the control-to-inference reformulation.","marker":"(Aström, 1970)"}],"fun_headline_variants":["UnHiPPO: Kalman-filtered init for noise-robust SSMs","HiPPO's uncertainty upgrade: UnHiPPO filters noise","Noise-aware SSM init from UnHiPPO's Kalman view","UnHiPPO: Posterior-inferring HiPPO for noisy data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation replaces the observed signal with its Legendre polynomial reconstruction, which is exactly correct only when the signal is a polynomial of the represented degree; for real-world noisy signals this substitution is an approximation, and the pseudo-inverse regularization used to stabilize the resulting dynamics is an additional heuristic choice.","fun_headline_variants_meta":{"raw":{"variants":["UnHiPPO: Kalman-filtered init for noise-robust SSMs","HiPPO's uncertainty upgrade: UnHiPPO filters noise","Noise-aware SSM init from UnHiPPO's Kalman view","UnHiPPO: Posterior-inferring HiPPO for noisy data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000592,"raw_usage":{"total_tokens":2764,"prompt_tokens":926,"completion_tokens":1838,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":1755}},"tokens_in":542,"tokens_out":1838,"duration_ms":15669,"temperature":1.0,"reasoning_tokens":1755,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:25:03.651851+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the UnHiPPO recurrence on a clean polynomial signal of degree at most $N-1$, where the Legendre projection is exact, with $\\sigma^2 \\to 0$: if the posterior mean does not track the known HiPPO projection coefficients, the substitution underlying the dynamics is wrong even in the noise-free case. On synthetic signals with known ground truth (e.g., Gaussian process samples with added Gaussian noise), sweep $\\sigma^2$ and compare the reconstruction error of UnHiPPO against HiPPO: if no value of $\\sigma^2$ yields a lower error, the claimed noise filtering does not occur.","supporting_citations":[],"review_version":1}