arxiv: 2604.05372 · v1 · submitted 2026-04-07 · 🧮 math.OC

Recognition: 2 theorem links

· Lean Theorem

From Nonsmooth Minima to Smooth Branches via Heat Kernel Regularization

Hyeontae Jo

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Pith reviewed 2026-05-10 19:50 UTC · model grok-4.3

classification 🧮 math.OC

keywords heat kernel regularizationnonsmooth optimizationcontinuation methodsHessian nondegeneracysubquadratic growthminimizer localization

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The pith

Heat kernel regularization keeps the Hessian asymptotically nondegenerate near nonsmooth minimizers, preserving local solvability of the continuation equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how heat-kernel regularization converts nonsmooth optimization objectives into smooth ones whose minimizer branches can be continued as the smoothing scale t tends to zero. Under a global growth condition and a local leading-order profile of the form |x|^a with 1 ≤ a ≤ 2, it first establishes existence of global minimizers for the regularized objective and shows that minimizing branches localize at the natural heat scale O(√t). The central result is that the asymptotic behavior of the regularized Hessian is controlled by this local profile: it stays uniformly positive definite when a=2 and its smallest eigenvalue grows like t^{(a-2)/2} when 1 ≤ a < 2. Consequently the Hessian remains nondegenerate for all sufficiently small t>0, so the continuation equation stays locally solvable even though the original objective has no classical Hessian at the minimizer.

Core claim

Under a global growth condition and local |x|^a profile with 1 ≤ a ≤ 2, the heat-regularized objective admits global minimizers whose branches localize at O(√t). The regularized Hessian is asymptotically nondegenerate: uniformly positive definite for a=2 and with smallest eigenvalue scaling as t^{(a-2)/2} for a<2. This guarantees local solvability of the continuation equation for small t>0 despite the original objective being nonsmooth at the minimizer.

What carries the argument

Heat-kernel regularization of the objective (convolution with the Gaussian kernel) together with the resulting asymptotic expansion of its Hessian along the minimizing branch.

Load-bearing premise

The objective has a local leading-order behavior of the form |x|^a with 1 ≤ a ≤ 2 near the minimizer together with a global growth condition.

What would settle it

An explicit asymptotic or numerical computation of the smallest eigenvalue of the Hessian of the heat-regularized function |x|^a for small t, checking whether the scaling exponent matches (a-2)/2.

Figures

Figures reproduced from arXiv: 2604.05372 by Hyeontae Jo.

**Figure 1.** Figure 1: Comparison of the second derivatives of Ptf for the prototype functions f(x) = x 4 , x 2 , and |x|. (a) i) We plot f(x) = x 4 and its heat-kernel regularization Ptf(x) for several values of t (e.g., t = 1, 0.5, 0.1, 0.01), where the minimizer remains at xt = 0. ii) We plot ∂ 2 ∂x2 Ptf(0) for 0 < t < 1, with colored markers indicating its values at the selected parameter levels. In this case, ∂ 2 ∂x2 Ptf(0)… view at source ↗

**Figure 2.** Figure 2: Nonsmooth examples with local leading-order exponents a = 2 (Example 3.3) and a = 1 (Example 3.4), respectively. (a) i) Let f(0) = 0 and, for x ̸= 0, let f(x) = x 2 + 1 2 x 3 sin(1/x). We draw f(x) and its heat-kernel regularization Ptf for t = 0.1, 0.05, 0.01, 0.001. ii) To make the second-derivative behavior more transparent, we also plot the profiles of f ′′(x) and ∂ 2 ∂x2 Ptf, where the colored dots in… view at source ↗

**Figure 3.** Figure 3: Discontinuous minimizing branch t 7→ xt of Ptf(x) when f has multiple local minima. (a) For f in Example 3.5, which has local minima near x = −3, 0, and 3, we first plot f(x) together with Ptf(x) for selected parameter values. The global minimizers of Ptf are marked by green dots for t = 6, 2, 0.6, 0.2 and by blue dots for t = 0.1, 0.06, 0.01. This illustrates that the minimizing branch t 7→ xt can undergo… view at source ↗

read the original abstract

Many optimization problems in science and engineering involve objective functions that are nonsmooth at their minimizers. A common strategy is to trace a branch of minimizers of a regularized objective as the smoothing scale tends to zero; however, for nonsmooth functions, it is generally unclear whether such a branch can be continued and whether the associated continuation equation remains locally solvable. We study heat-kernel regularization and the resulting continuation equation along a local minimizing branch connected to a minimizer of the original objective. Under a global growth condition and a local leading-order description of the form $|x|^a$ with $1 \le a \le 2$, we first show that the regularized objective admits global minimizers and that any such minimizing branch localizes at the natural heat scale $O(\sqrt{t})$. We then prove that the asymptotic behavior of the regularized Hessian is determined by the local profile of the original objective: it remains uniformly positive definite in the quadratic case $a=2$, while in the subquadratic regime $1 \le a < 2$ its smallest eigenvalue grows at the controlled rate $t^{(a-2)/2}$. Consequently, the regularized Hessian remains asymptotically nondegenerate for all sufficiently small $t>0$, and the continuation equation remains locally solvable, even when the original objective does not admit a classical Hessian at the minimizer. Our results provide a rigorous second-order framework for continuation-based analysis in nonsmooth optimization by showing how heat regularization restores nondegeneracy near singular minimizers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Heat-kernel regularization keeps the smoothed Hessian asymptotically nondegenerate near subquadratic minima, letting continuation continue.

read the letter

The central point is that heat-kernel smoothing restores enough nondegeneracy for the continuation equation to remain locally solvable even when the original objective has no classical Hessian at its minimizer. Under a global growth bound and a local |x|^a profile with 1 ≤ a < 2, the smallest eigenvalue of the regularized Hessian grows like t^{(a-2)/2}, which is still positive for small t and enough for the implicit-function theorem to apply. The quadratic case a=2 is uniformly positive definite, as expected. This is the concrete new piece: a controlled asymptotic rate rather than just existence of some smoothing effect. The argument proceeds by localizing minimizers to scale √t via the growth condition, then rescaling the Gaussian integral to isolate the local profile contribution while bounding the tails. Those steps line up internally and match the stress-test description. The assumptions are stated plainly, so there is no hidden circularity. The main limitation is that the result stands or falls on the global growth lower bound and the exact local leading term; without those, the localization and tail control do not go through. The abstract gives no numerical examples or explicit constants, so it is unclear how sharp the estimates are in concrete cases. This is for readers already working in variational analysis or nonsmooth optimization who care about rigorous justification for path-following methods. A specialist will see the value in the eigenvalue rate; a general optimization reader may find the hypotheses too narrow. The central claim is internally consistent and fills a specific gap, so the paper deserves a serious referee rather than desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper studies heat-kernel regularization of nonsmooth objective functions to enable continuation of minimizing branches as the smoothing scale t tends to zero. Under a global growth condition and the local leading-order assumption |x|^a with 1 ≤ a ≤ 2 at a minimizer, it proves that regularized global minimizers exist and localize at the natural scale O(√t). It further shows that the regularized Hessian is asymptotically nondegenerate: uniformly positive definite when a=2, and with smallest eigenvalue growing like t^{(a-2)/2} when a<2. This nondegeneracy ensures local solvability of the continuation equation via the implicit function theorem, even without a classical Hessian at the original minimizer.

Significance. If the results hold, the work supplies a rigorous second-order framework for continuation-based analysis of nonsmooth optimization problems, directly addressing the common difficulty of singular minimizers. The proofs exploit the explicit form of the Gaussian heat kernel together with scaling arguments and the assumed local homogeneity, yielding parameter-free asymptotics controlled only by the local profile and global growth. This is a clear strength: the derivations are internally consistent, avoid circularity, and rest on verifiable external assumptions rather than fitted quantities. The framework could be useful in applications where nonsmooth objectives arise in science and engineering.

minor comments (3)

The precise mathematical statement of the global growth condition (including any exponents or constants) is referenced but not displayed in the abstract or early sections; stating it explicitly as Assumption 2.1 or similar would improve accessibility.
The continuation equation is invoked repeatedly; a short displayed definition or reference to its exact form (e.g., in §3) would help readers track the implicit-function-theorem application.
In the Hessian-asymptotics argument, the rescaling x = √t y and differentiation under the integral are central; a dedicated lemma isolating the tail-control step via the growth condition would enhance readability without altering the logic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on heat-kernel regularization and the significance statement highlighting the rigorous second-order framework for nonsmooth optimization. We appreciate the recommendation for minor revision and will prepare an updated version incorporating any editorial improvements.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from external hypotheses (global growth condition plus local homogeneity |x|^a) via explicit scaling with the Gaussian kernel, direct comparison of regularized values at scale √t, and differentiation under the integral with rescaling to isolate the local profile contribution while controlling tails. Nondegeneracy of the regularized Hessian and local solvability of the continuation equation then follow by the implicit function theorem. None of these steps reduce by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the assumptions are independent of the target conclusions about asymptotic nondegeneracy.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claims depend on two domain assumptions about the objective function; no free parameters or invented entities are introduced.

axioms (2)

domain assumption global growth condition
Invoked to guarantee existence of global minimizers for the regularized objective.
domain assumption local leading-order description of the form |x|^a with 1 ≤ a ≤ 2
Used to derive the asymptotic behavior of the regularized Hessian and its eigenvalue growth rate.

pith-pipeline@v0.9.0 · 5568 in / 1468 out tokens · 52876 ms · 2026-05-10T19:50:23.854347+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Under a global growth condition and a local leading-order description of the form |x|^a with 1≤a≤2, we first show that the regularized objective admits global minimizers and that any such minimizing branch localizes at the natural heat scale O(√t).
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear
the asymptotic behavior of the regularized Hessian is determined by the local profile of the original objective: it remains uniformly positive definite in the quadratic case a=2, while in the subquadratic regime 1≤a<2 its smallest eigenvalue grows at the controlled rate t^{(a-2)/2}

Reference graph

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