Yuda Bi

The question

Excess kurtosis, the standardized fourth cumulant, is a central contrast of ICA and the basis of FastICA’s kurtosis-type nonlinearity, and it is known to fade as mixtures widen: the central limit theorem pushes wide projections toward Gaussianity. Earlier analyses took contrast strength as given. This letter measures how much contrast survives, at the population level, when a projection spreads its weight over many sources. That is the regime of high-model-order group ICA in neuroimaging, where more samples cannot restore contrast that the mixture itself has removed.

Results

• Theorem 1 (Sharp redundancy bound). For any standardized projection y = Σ_j w_js_j, |κ(y)| ≤ κ_max Σ_j w⁴_j = κ_max/R_eff (Corollary 1). Under balanced mixing (max_j w²_j ≤ c_b/R) this is at most c_bκ_max/R, and equal weights with equal kurtoses attain |κ₀|/R exactly. Proposition 1 shows that balance is generic inside well-conditioned blocks, with c_b growing at most logarithmically in R.
• Corollary 2 (Necessary model-order screening condition). The contrast of a balanced projection can exceed the estimation noise floor σ₀/√T only if R < c_bκ_max√T / σ₀, so doubling the tolerable width takes four times the data. The letter gives a coarse recipe that estimates this ceiling from data and flags hopeless model orders.
• Theorem 2 (Purification lower bound). Keeping m ≪ R sources of one kurtosis sign and renormalizing their weights gives |κ(y_pur)| ≥ κ_min/m, with κ_min the smallest kurtosis magnitude among the kept sources, whatever R is. A data-driven version picks the subset from a preliminary separation, without an oracle.

Evidence

Balanced mixtures of Student-t sources follow |κ̂(y)| ≈ 1.57/R (R² = 0.994), and the estimator noise crosses the population contrast near T ≈ 3 × 10⁴, as the screen predicts. Keeping m = 5 sign-consistent sources holds the contrast near 0.52 (oracle) and 0.59 (data-driven) from R = 20 to 200, while the full mixture falls from about 0.10 to 0.03.

On resting-state fMRI from COBRE (n = 155), the per-subject kurtosis gap between the leading and the typical component time courses shrinks from model order 53 to 100 (paired one-sided Wilcoxon test, p = 1.74 × 10⁻²⁷). The letter presents this as a supportive sanity check rather than a controlled test, since changing the model order also changes the signal subspace.

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The question

In overcomplete ICA, p sensors observe q > p independent sources. Pairwise non-collinear mixing directions are still identifiable for non-Gaussian sources, but two questions were open. How many samples does the problem need, for any estimator? And can the direction of one source of interest be estimated without modeling the others? The letter answers both from the product identity above.

Results

• Theorem 1 (Aliasing barrier). With two sensors and q equally spaced sources sharing a smooth symmetric law (Gram–Charlier laws, or binary symbols in Gaussian noise), a rotation by π/(2q) leaves every cumulant of order below 2q unchanged and moves the observation law exponentially little in total variation. Resolving the directions therefore takes e^Θ(q) samples, necessary for every estimator and sufficient for a family of sources; for the Gram–Charlier law φ₄, 1.5288q − O(log q) ≤ log n^*(q) ≤ 2.3689q + O(1). Already at q = 10 the squared Hellinger distance is 4.1 × 10⁻⁹, so at least 6 × 10⁷ samples are needed.
• Theorem 2 (Immunity). If the characteristic function of a target source has a real zero, as for equiprobable binary or pulse-amplitude symbols, complex-exponential moment conditions on its mixing direction are exactly unbiased, whatever the number, strength, laws, and mutual dependence of the other sources, provided they are independent of the target. Each such source leaves zero lines in the observed characteristic function that no other source can remove (Corollary 1); the zero-line estimator CFZ locates the target’s line within a prior window that contains no other source with such zeros.
• Propositions 1 and 2 (Moments are immune, variance is not; Price of local immunity). Immunity covers the moments, not the variance: for complex-exponential (Fourier) moments, each added interferer multiplies the Jacobian of the moment conditions by a factor of modulus at most one. Locally, an estimator that must stay unbiased against arbitrary weak interference keeps a computable share of the information: zero for Gaussian and Laplace sources, above 90% for binary symbols in Gaussian noise at 10 dB.

Evidence

The simulations estimate the direction of a binary target at 17°, seen by a two-axis vector sensor at 10 dB among Gaussian, four-level PAM, impulsive (Bernoulli–Gaussian), and heavy-tailed Student-t interferers. At n = 10⁵ snapshots the RMSE of CFZ is 0.15°–0.40° across seven settings, its standard deviation within 10% of the prediction of Proposition 1, with no error above 3°.

Baselines with a fixed interference model level off instead: semi-blind likelihood with a Gaussian model (SB-G) and a kurtosis contrast under mixed or heavy-tailed interference, and a Student-t model (SB-T) under mixed interference at the lower power, with errors of up to 8.4°. A Gaussian-mixture likelihood matches or beats CFZ at one to two orders of magnitude more computation: where a model is adequate, likelihood is the more efficient route; immunity is what remains when none is.

Table II of the paper: direction error at n = 10⁵

Setting	CRB	CFZ [Prop. 1]	SB-G	SB-T	SB-GMM	KURT
G3 (.5)	0.22	0.282 [0.298]	0.28	0.32	0.28	0.33
A3 (.5)	0.20	0.336 [0.353]	0.26	0.41	0.30	0.37
M3 (.5)	0.14	0.204 [0.226]	5.5†	0.21	0.18	8.4†
M3 (.25)	0.12	0.147 [0.148]	1.4†	0.56†	0.13	7.1†
M6 (.25)	0.18	0.254 [0.237]	2.7†	0.40†	0.25	6.9†
M9 (.25)	0.25	0.396 [0.379]	3.6†	1.5†	0.37	6.3†
H3 (.25)	0.22	0.302 [0.312]	6.4†	0.29	0.28	6.3†

Root-mean-square direction error in degrees, with the CFZ standard deviation predicted by Proposition 1 in brackets. Interference is Gaussian (G), PAM-4 (A), mixed (M), or heavy-tailed (H); the digit is the number of interferers and the number in parentheses their power. † Absolute bias above a third of the RMSE and three standard errors, or more than 5% of absolute errors above 3°.

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The question

Some pipelines normalize an ICA criterion with statistics from records other than the ones that score it. FastICA’s sampleSize option scores a random fraction of the records while centering and whitening on all of them; real-time EEG pipelines keep the sphere of a calibration run; cross-fitting and reliability analyses refit on split or resampled records. The paper asks what this normalization mismatch costs, exactly, for one-unit and for symmetric fits; whether the cost can be removed without abandoning the protocol; and what agreement among differently normalized fits says about accuracy.

Results

• Theorem 1 (Local three-role influence law). Normalization is not orthogonal at first order, unlike the source density and smooth score parameters (Remark 1). For a one-unit fit with arbitrary scoring, scale, and centering weights, the influence function splits into score exposure, a scale channel, and a location channel, which gives the exact first-order cost above. Equal-size cross-fitting costs nothing; 10–20% reference or calibration normalizers cost up to hundreds of times the common-sample variance for log-cosh on sub-Gaussian targets.
• Theorem 3 (Symmetric fits under mismatch). For symmetric FastICA the law is pairwise. A pair pays through the difference of its sign-corrected scale weights, large for opposite curvature signs under log-cosh, and through the whitening’s own effective sample size, even when the roles match.
• Theorems 2 and 4 (Orthogonalized contrast; Orthogonalized symmetric contrast). Subtracting consistently estimated linear and quadratic terms removes the mismatch channels for one-unit and symmetric fits and leaves matched-role fits unchanged. What remains is the price of scoring fewer records and, for symmetric fits, the whitening’s effective sample size. Balanced reuse of the records reaches the common-sample law without any source moments (Corollary 2).
• Corollaries 3, 4, and 6 (Two-point targets; Exponential windows; Asymptotic localized confidence cone). For two-point targets such as BPSK, mismatch changes the rate from exact recovery to N^−1/2. With exponentially weighted roles, the whitening memory should be the longer one. No remedy reduces the common sampling error, so agreement among differently normalized fits on the same scored records is blind to it; the paper gives an observable confidence cone to report instead.

Evidence

Five real signals (an electrocardiogram from the MIT-BIH Arrhythmia Database, two natural images, and the horizontal gradients of two more) are mixed by known random matrices. Under 10% reference or 20% calibration whitening, plain one-unit refits pay 9.9 to 728 times the common-sample variance and plain symmetric fits 22 to 71 times their common-sample loss, in line with the first-order predictions. Orthogonalized one-unit refits pay 1.01–1.09 and 1.23–1.37 times it, against the predicted 1 and 1.25. Across the reference, calibration, and subsample protocols, the symmetric remedy cuts every loss by 6 to 14 times.

In a line-by-line port of the FastICA sampleSize code path with a 10% subsample, the default deflation mode returned no component in any of 100 runs, and the redrawn-subset symmetric iteration with tanh met its stopping test in only 5 of 100. Patching four lines with the orthogonalized nonlinearity brought the tanh losses to the predicted floor and its stopping rate to 100 of 100. With pow3, the package’s default nonlinearity, the patch also brings the fixed-subset losses close to the floor, but the redrawn iteration stops in only 13 of 100 runs.

Table II of the paper: one-unit refits on real signals

Signal	Kurtosis μ4	cg/vg	Reference, 10%		Calibration, 20%
			plain [prediction]	orth.	plain [prediction]	orth.
ECG	10.0	4.6	48.8298 [42.2]	1.04	30.7 [29.9]	1.31
Image (china)	1.5	59.2	728.1 [533.6]	1.03	451.8 [371.1]	1.26
Image (face)	2.2	25.5	201.3253 [230.6]	1.06	147.8274 [160.7]	1.37
Gradient (ascent)	19.6	2.9	28.6 [27.1]	1.09	19.3 [19.4]	1.28
Gradient (flower)	58.0	1.3	14.2 [12.9]	1.01	9.93 [9.5]	1.23

Mean loss in units of the common-sample variance (d = 5, N = 40 000, 300 replications, log-cosh, independently resampled records), with the first-order prediction for plain refits in brackets; orthogonalized refits are predicted at 1 and 1.25. Reference: a random 10% normalizes and all records are scored. Calibration: the first 20% normalizes and the rest are scored. Superscripts give the number of returned roots when fewer than 300.

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The question

An ICA run returns a demixing matrix but no measure of its accuracy. Since the sources are unknown, the natural source-free diagnostic is how well the estimate diagonalizes an observed statistic; for the fourth-order cumulant tensor, that is the off-diagonal residual. Upper bounds turn the residual into an error bar for one estimate, but deciding which of two estimates is closer to the sources also needs a computable lower bound. The question is practical in template-guided pipelines, where a guided estimate matches its template better than a data-only fit by construction, which says nothing about its distance to the sources.

Results

• Theorem 1 (Optimal global reverse bound). For every orthogonal candidate B and cumulants of any sign, r(B) ≤ √2 Γ D(B), where D is the distance to the signed-permutation orbit of the sources and Γ the root-sum-square of the two largest cumulant magnitudes. No smaller constant holds for all candidates (whenever the cumulant tensor is nonzero), and the bound improves on the column-wise constant by a factor of up to √d.
• Theorem 2 (Finite, locally sharp upper bound). If γ > 0 and the candidate is known to satisfy D(B) ≤ D̄ < √2, then D(B) ≤ K(η) r(B)/γ with η = 1 − D̄²/2; the coefficient decreases to the optimal local value 1/√2 as D̄ → 0.
• Theorem 3 (Observable bracket). A spectral estimate of Γ from the d² × d² unfolding of the observed tensor makes both sides computable under a valid tensor-error radius δ, giving L(B) ≤ D(B) ≤ C(B). With an exact tensor (δ = 0) the bracket is asymptotically exact near the source orbit for equal-magnitude cumulants (Corollary 1), and C(B) < L(V) certifies that B is closer to the sources than V.

Evidence

In a paired ablation with the same tensors, radii, and candidates, the new bounds raise the certified share of candidates truly closer than their reference from 22.6% to 45.0%, with no false verdicts. Median looseness C/D falls from 8.55 to 6.06, and across 3,360 exact-tensor comparisons the certified count rises from 132 to 1,253, again with no false claims.

The paper is explicit about the limits. Certifying more comparisons does not by itself lower the error of the returned estimate; no guided estimate can be certified closer to the sources than a residual-minimizing data-only fit; and all experiments are synthetic, with validation on real fMRI data left to future work.

Table 1 of the paper: fixed-candidate ablation

Variant	E1 C/D	E3 claims	Sensitivity (%)	Final/base
Legacy bounds	8.55	120	22.6	1.000
New lower bound	8.55	205	38.6	1.000
New upper bound	6.06	130	24.5	0.910
Both	6.06	239	45.0	0.910

E1 looseness C/D and E3 final/base error are medians; claims count C < L before the decision rule; sensitivity is the certified share of the 531 E3 candidates that are truly better than their reference. Final/base uses the tolerance τ/G_d = 0.01.

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The question

A stationary coordinate whose autocorrelation is a single exponential looks like an equilibrium Ornstein–Uhlenbeck (OU) process: an equilibrium OU model reproduces the complete Gaussian OU path law even when the hidden dynamics dissipates. A dissipation bound that uses only such a record must therefore allow zero, and bounds built on short-time correlations, correlation times, or cross-correlation asymmetry all vanish for an exact Gaussian OU record. The paper asks what thermodynamics survives inside this blind spot once the heat exchanged with the observed bath is measured independently.

Results

• Curvature bound (Eq. 10). For stationary overdamped diffusions with constant diffusion, independent observed and hidden noises, even coordinates, nonlinear forces, and any finite number of hidden coordinates, σ ≥ q²/(2q + Δ_C/D_X) with q = Q_X/(k_BT_X), whenever the denominator is positive. It is the short-time thermodynamic uncertainty relation for the current weighted by the deviation of the observed drift from its variance-matched OU drift, which is optimal among combinations of drift and displacement; the variance sum rule gives its variance. The bound is strict at nonzero heat and a sharp infimum for every allowed heat and curvature defect.
• Zero curvature defect (Eq. 11). At Δ_C = 0, which every exponential autocorrelation satisfies, heat can flow only into the observed bath, and positive heat gives σ > Q_X/(2k_BT_X). Neither Gaussianity nor a Markov observation law is needed.
• Two-temperature threshold (Eqs. 12–14). With conservative forces, constant positive mobilities, no external work, and one common hidden temperature, σ = q(1 − T_X/T_H), so at Δ_C = 0 and Q_X > 0, T_H > 2T_X: the coordinate with the exponential record is the colder one, whatever the nonlinear forces and however many hidden coordinates there are. With several hidden baths that all supply heat, the heat-weighted harmonic mean of their temperatures replaces T_H; the hottest bath is constrained, the coldest is not.
• Reciprocal models and finite bandwidth (Eqs. 16–20). A reciprocal two-coordinate model realizes every T_H/T_X > 2 at fixed observed law and heat, approaching the threshold only as its hidden relaxation rate diverges. In linear conservative networks of any size with one hidden temperature, an exactly exponential record with positive heat couples to exactly two relaxation modes, so a cap Λ_max on the relaxation rates raises the threshold to 2 + (2a + q)/(Λ_max − a − q).
• What response and finite data add (Eqs. 22–27). At Δ_C = 0 the heat is q = −G′(0⁺) − a, the initial slope of the response to a small displacement, which the passive record does not contain. Conversely, a five-dimensional OU diffusion whose sampled covariance is exactly exponential at every lag can have a curvature defect of either sign and any size, so certifying Δ_C = 0 requires a bandwidth or model assumption.

Evidence

A harmonically trapped particle driven by a programmed OU force makes the blind spot concrete. At the tuned stiffness the position spectrum is exactly that of an OU process at the force rate β, so the variance and corner-frequency calibrations agree with each other while both underestimate the stiffness; yet the probe delivers heat Q_X = k_BT_X(α − β) > 0 to its bath. With the drive treated as an even coordinate, the entropy production equals this heat in units of k_BT_X, and the sharp lower limit is half of it.

The heat must be measured independently, for example through a calibrated response measurement, and the paper quantifies how well conditioned that is. In a simulation that gives the estimator the exact two-coordinate model (q = 2, σ = 5/3), with 250 passive/driven pairs per frequency of 65,537 exactly sampled points each, the single-frequency inverse returns a physical solution in 99.2% of trials at ω = 4 but recovers the heat within 25% in only 66.8% of them; at ω = 0.05 and 64 these fractions fall to 55.6% and 32.0%. The simulation contains intrinsic fluctuations only; sensor noise and calibration error would add to them. Near the tuned stiffness, the statistical evidence against an OU law vanishes quadratically in the detuning while the heat stays finite.

For an experimental test the paper proposes a temperature scan in coupled thermal RC circuits. With a fixed drift (α = 2/τ, β = 5/τ, g = m = 2/τ), the cold coordinate becomes exactly exponential at T_Y/T_X = 11/4 while heat keeps flowing; with 1 MΩ resistors and τ = 10 ms the capacitors are 5 nF and 10/3 nF. No experiment is reported.

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The question

Two harmonically trapped probes, each in its own thermal bath, are driven by one persistent, unobserved OU force. The drive can be tuned so that each position record has exactly the statistics of an equilibrium particle, which defeats every single-channel method; at equal response rates the joint record is also exactly time reversible, which defeats every method based on the arrow of time. The paper asks which response rates and probe–bath heat rates can still be identified from the two passive records.

Results

• Two-probe sampled inverse (Eqs. 8–11). At double cancellation (both probes tuned, nonzero drive couplings), each sampled record is a first-order autoregression with coefficient b = e^−βΔ. Filtering both records with e_n = X(nΔ) − b X((n−1)Δ) gives cross-covariance tails proportional to a^k−1₁ at positive lags and a^k−1₂ at negative lags, so consecutive lag ratios determine both response rates, equal responses included. The amplitudes and the zero-lag value are given in closed form.
• Heat without a time arrow (Eqs. 14–21). The Harada–Sasa relation converts each recovered rate into heat. At equal responses the summed heat in thermal units, 2(α − β), is positive while the observed path irreversibility Π₁₂ is exactly zero: the relative coordinate rejects the common force and relaxes at the response rate. Time-symmetric kinetic bounds are positive in this reversible pair but bound rather than attribute the heat (at (α, β) = (3, 1) the bound is 1.6 and the summed heat rate 4), and near equal response Π₁₂ vanishes quadratically while the heat stays finite.
• Fisher information (Eqs. 23–26). In the ideal continuous experiment with known diffusion, and with all other parameters estimated jointly, the information one probe carries about its own response rate vanishes quadratically with the weight of the response mode in its covariance, whereas the joint information matrix of the two probes is positive definite at every cancellation point with nonzero forcing, including equal responses. At (α₁, α₂, β) = (1, 2, 0.5) the joint values per unit time are 0.3582 and 0.01425, against zero for each probe alone. As the forcing vanishes at equal response, with heat rate h, the relative variance of an individual bath heat scales, to leading order, as 2β³/(h⁴T_obs) and that of the summed heat as 3β/(h²T_obs), for record duration T_obs.
• Why bath independence is needed (Appendix C 4). With correlated probe noise and a partly shared drive, a fine-tuned family keeps the complete two-channel law while its response rate varies, so the response disappears from the data. An independent response measurement, switching off the drive at unchanged stiffness, or calibrating the drive intensity resolves it.

Evidence

Exact-sample likelihood fits on synthetic records, which are given neither cancellation nor the drive amplitude, measure finite-record precision. Across nine designs of 100 stationary pairs (N = 2000, rates (1, 2, 0.5)), neither the cancellation-constrained covariance fit nor the general Kalman likelihood dominates; all 900 fits pass the interval-validity checks, and nominal 90% intervals for the response rates cover 83–94%, the low end not significant after a Holm adjustment.

At equal responses (1, 1), with unknown and unequal diffusivities and means, 120 records give heat-rate RMSE of 0.12 and 0.15 for true rates of 0.5, and the heat intervals cover 109 and 107 of 120. Across this and a near-equal design (rates 0.95 and 1.05), the median interval widths are 90–94% of the fitted heat: at this duration the heat is identified, not yet precise.

The paper proposes a two-trap optical test: spheres of radius 0.5 µm at 300 K with α₁ = 100 s⁻¹, α₂ = 200 s⁻¹, and β = 50 s⁻¹, rms forces of 0.0765 and 0.1711 pN, and bath heat rates of 50 and 150 k_BT s⁻¹. For an equal-response configuration (α = 100 s⁻¹, β = 50 s⁻¹, 5 ms sampling), a local asymptotic budget for ideal sampling asks for at least about 19 s of data to reach 20% precision on an individual bath heat, and about 22 min at weak forcing (6 s and 62 s for the summed heat); these durations are necessary, not sufficient. All numerical results use synthetic data; no experiment is reported.

Table I of the paper: paired recovery at cancellation

Δ	Tobs	Cancellation covariance fit	General Kalman likelihood
0.1	200	(0.173, 0.845)	(0.124, 0.708)
0.5	1000	(0.085, 0.309)	(0.060, 0.345)
1	2000	(0.059, 0.234)	(0.056, 0.267)
2	4000	(0.062, 0.373)	(0.088, 0.496)

Unconditional RMSE of the response rates (α₁, α₂), in inverse time, from 100 stationary pairs per row with N = 2000 exact transitions and (α₁, α₂, β) = (1, 2, 0.5). The covariance fit imposes cancellation; the Kalman likelihood estimates all seven covariance parameters and profiles both means without that constraint.

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