Beyond the Semicircle: Free Diffusion Models with Prescribed Equilibria

Swagatam Das

math.PROct 26, 2025

Beyond the Semicircle: Free Diffusion Models with Prescribed Equilibria

Authors: Swagatam Das

Abstract

A growing class of machine-learning objects -- covariance and Gram matrices, kernel and attention matrices, MIMO channel matrices, density operators -- are naturally spectra rather than coordinate vectors. Building a denoising diffusion model for such data by corrupting eigenvalues coordinatewise is not merely elegant: it converges to the wrong limit, because eigenvalues repel rather than move independently. Free probability theory supplies the correct forward process, with free convolution replacing classical convolution and Voiculescu's conjugate variable replacing the score, but constant-coefficient free diffusions have a hidden limitation of their own: their only possible equilibrium is the semicircular law, whatever the target distribution looks like. We show that state-dependent free volatility removes this restriction. For any sufficiently regular compactly supported target law, we explicitly construct, through a closed-form Hilbert-transform drift, a free Fokker--Planck flow having that law as a stationary spectral distribution. Under an additional convex-potential condition, the construction admits a globally relaxing diffusion interpretation, reverse-time dynamics, and a trainable score. The generalization is genuine rather than cosmetic: no change of spectral variable reduces it to the constant-coefficient case, and it is a Wasserstein gradient flow only for an explicitly characterized family of coefficients that contains no bounded non-constant member. We instantiate the construction on the Marchenko--Pastur law, the limiting spectrum of isotropic sample covariance matrices, and validate it numerically: the free-versus-coordinatewise gap, an exact dimension-independent score driving reverse-time matrix dynamics at several matrix sizes, the designed Marchenko--Pastur equilibrium, and end-to-end generation with a learned score.

Figures & tables

Figure 1: Three ways to corrupt a spectrum, and what each can produce as an equilibrium. Left: coordinatewise noise gives the wrong limit entirely. Middle: the correct (free) process, but only ever semicircular. Right: this paper – an arbitrary designed target.

Classical (vectors)	Free (spectra)
Gaussian convolution	free convolution $\boxplus\gamma_{v}$
score $-\nabla\log p_{t}$	conjugate variable $\xi_{\mu_{t}}=2H\mu_{t}$
Tweedie, denoising score matching	free Tweedie identity (App. D )
reverse SDE	reverse free SDE ( 4 )
constant-coefficient limit: Gaussian	constant-coefficient limit: semicircle only
Langevin drift $\tfrac{1}{2}\nabla\log p_{*}$	$b_{}=-H\mu_{}=-\tfrac{1}{2}\xi_{\mu_{*}}$ ( $f\equiv 1$ )

Table 1: Dictionary between classical and free spectral diffusion.

Figure 2: The equilibrium-design construction: a target law $\mu_{*}$ determines, through a single Hilbert-transform formula, a drift $b_{*}$ for which $\mu_{*}$ itself is a stationary spectral law of the free flow.

Figure 3: Real S&P 500 correlation spectrum ( $N=100$ , $T=1258$ ) against the Marchenko–Pastur bulk prediction and the best-fitting semicircular law – the only shape a constant-coefficient free diffusion can produce.

$v$	$0.25$	$1.00$	$2.56$
free ( $\boxplus$ )	0.0010	0.0013	0.0022
coordinatewise ( $*$ )	0.109	0.219	0.275

Table 2:

W_{1}

distance to the empirical spectrum. Free tracks the finite-

N

fluctuation; coordinatewise is off by two orders of magnitude.

Figure 4: Reverse-time matrix dynamics driven by the exact free score of the two-atom law at $N=50,200,600$ against the target density, and $W_{1}$ error against $N$ .

score	learned	oracle	coordinatewise
$W_{1}$	0.073	0.062	0.121

Table 3:

W_{1}

(generated, data), same reverse-time sampler, three scores.

Figure 5: Left: learned, exact, and coordinatewise scores at $\alpha_{t}=0.45$ . Right: spectra generated by each, against the data spectrum.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Reverse-time spectral flow from the semicircular law back to the two-atom law. Histograms: $6\times 10^{4}$ trajectories. Curves: target density at the corresponding time.

Figure 7: Corrupting a Hermitian matrix with Hermitian noise convolves its spectrum freely, not classically. Grey: eigenvalues of $X_{0}+\sqrt{v}\,G$ , $N=2500$ , GUE noise. Blue: free prediction. Red dashed: coordinatewise prediction, with Gaussian tails where the truth has compact support.

Figure 8: Forward free OU (Dyson) flow from the two-atom law. Histograms: a single Hermitian-diffusion realization, $N=1200$ . Curves: the free Fokker–Planck prediction. The spectrum passes bimodal to unimodal, converging to semicircular.

Figure 9: Phase diagram of the two-atom support transition. Color: density at the gap center of $\tfrac{1}{2}(\delta_{-a}+\delta_{a})\boxplus\gamma_{v}$ . White: numerically detected boundary. Red dashed: the exact transition $v^{*}=a^{2}$ of § 6 , confirmed as an identity in $a$ , not a single point.

Figure 10: Forward free OU flow from a spiked covariance spectrum, $N=600$ . The three outliers merge into the bulk as the flow proceeds – the finite-rank phenomenon that Theorem 4.8 does not cover (Appendix H ).

Figure 11: Convergence to the free limit, log-log, averaged over independent realizations. Left: $W_{1}$ against the exact free marginal, $N^{-1}$ reference. Right: the same comparison against a finite Monte Carlo sample instead, $N^{-1/2}$ reference – the apparent rate is that of the reference sample, not of the matrix model; see § 6 .

Figure 12: Forward flow in spacetime. Left: density $\psi_{t}(x)$ over $(\Lambda(t),x)$ with support/gap edges from the free theory and the exact transition time $\Lambda^{*}$ . Right: an $N=40$ eigenvalue trajectory against the same edges.

Figure 13: Left: the LSI, Talagrand and HWI ratios, all bounded by $1$ . Middle: exponential decay of $\mathcal{D}$ against $e^{-\Lambda(t)}$ . Right: pointwise de Bruijn identity $-\mathrm{d}\mathcal{D}/\mathrm{d}\Lambda=\tfrac{1}{2}I$ .

Figure 14: Reverse-time free flow reconstructing a spiked covariance spectrum from the semicircular law, exact score. Bulk and the three outliers are both recovered.

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Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the vocabulary size

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Abstract

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