m-Sharpness: Multi-Domain Definitions and Applications

• m-Sharpness is a polysemous term that refines conventional sharpness by optimizing over equivalence classes in neural networks, percolation, and quantum measurements.
• In neural network contexts, it is used to achieve scale invariance via minimizing Hessian traces or by evaluating sharpness on micro-batches (SAM/mSAM) to enhance generalization.
• Beyond deep learning, m-Sharpness appears in percolation theory for multiscale estimates and in quantum information as a basis-independent measure of measurement sharpness.

m-Sharpness is a polysemous technical term whose meaning depends on the research tradition in which it is used. In the cited literature, it denotes at least four distinct constructions: Minimum Sharpness, a scale-invariant Hessian-trace measure for neural networks; $m$-sharpness in Sharpness-Aware Minimization (SAM), where sharpness is evaluated on subsets or micro-batches rather than on the full training set; a multiscale sharpness estimate in Bernoulli percolation indexed by a mesoscopic scale $m$; and a basis-independent notion of measurement sharpness obtained by maximizing POVM coherence over all reference bases (Ibayashi et al., 2021, Foret et al., 2020, Vanneuville, 2022, Baek et al., 7 Jul 2026). In all of these settings, the central issue is not merely whether an object is “sharp,” but which perturbations, equivalence classes, or scales ought to define the relevant notion of sharpness.

1. Terminological scope and major usages

The symbol $m$ does not carry a single universal meaning across the literature. In Minimum Sharpness, it encodes a minimum over scale-equivalent parameterizations. In sharpness-aware optimization, it indexes mini-batch or micro-batch size. In percolation, it is a mesoscopic scale parameter controlling a multiscale estimate. In the measurement-theoretic setting, the term is used for maximal coherence under basis optimization, not for a batch or scale parameter.

Usage of m-Sharpness	Core object	Representative papers
Minimum Sharpness	$$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$	(Ibayashi et al., 2021)
$m$-sharpness in SAM/mSAM	Sharpness computed on size-$m$ subsets or micro-batches	(Foret et al., 2020, Behdin et al., 2022, Behdin et al., 2023, Luo et al., 22 Sep 2025)
Multiscale percolation sharpness	$\theta_{2n}(p-2\theta_m(p))$ estimate	(Vanneuville, 2022)
Measurement-theoretic m-Sharpness	$$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$	(Baek et al., 7 Jul 2026)

This plurality matters because claims about “m-Sharpness” are not interchangeable across domains. The deep-learning uses concern curvature, local perturbation geometry, and implicit bias; the percolation use concerns phase-transition sharpness; the POVM use concerns resource-theoretic distance from trivial measurements (Ibayashi et al., 2021, Behdin et al., 2022, Vanneuville, 2022, Baek et al., 7 Jul 2026).

2. Minimum Sharpness as a scale-invariant neural-network quantity

Minimum Sharpness, or m-Sharpness in the 2021 formulation, was introduced to resolve the scale-sensitivity of ordinary Hessian-based sharpness. For positively homogeneous networks, one may rescale layers without changing the realized function, yet the Hessian trace may change arbitrarily. The construction therefore replaces sharpness at one parameter vector by the minimum sharpness over its scale-equivalence class (Ibayashi et al., 2021).

For a feedforward network

$$f(x\mid \theta) = \mathbf{W}_{D}\phi\!\left( \mathbf{W}_{D-1}\phi \cdots \phi(\mathbf{W}_{1}x)\cdots \right),$$

the layerwise scale transformation is

$$\theta'=\alpha(\theta)=\{\mathbf{W}_d'=\alpha_d\mathbf{W}_d\}_{d=1}^D, \qquad \alpha_d>0,\quad \prod_{d=1}^D \alpha_d=1.$$

Under this transformation,

$m$0

Minimum Sharpness is then defined as

$m$1

For no-bias FCNNs with non-negative homogeneous activations, the minimization admits a tractable layerwise form: $m$2 where $m$3 is the Hessian block for layer $m$4. By the arithmetic–geometric mean inequality, this yields the closed form

$m$5

The resulting quantity is invariant to the admissible scaling action: $m$6

The conceptual shift is substantial. Ordinary sharpness is parameterization-dependent; Minimum Sharpness is intended to be a property of the represented function modulo the scale symmetries induced by positive homogeneity. The paper also argues that the same logic extends beyond no-bias FCNNs to architectures with bias terms and convolutional layers, with the appropriate scaling factors inserted (Ibayashi et al., 2021).

3. Exact computation and empirical behavior of Minimum Sharpness

The practical obstacle for Hessian-based quantities is the cost of forming or approximating the full Hessian. The Minimum Sharpness paper addresses this by deriving an exact decomposition of the Hessian trace for softmax classification that avoids explicit Hessian construction (Ibayashi et al., 2021).

The decomposition is

$m$7

with

$m$8

where $m$9 is the logit for class $m$0, $m$1, and $m$2. The layerwise decomposition

$m$3

then interfaces directly with the geometric-mean formula for $m$4.

The practical consequence stated in the paper is that computation requires only $m$5 forward/backward passes per batch, rather than explicit Hessian formation. In the reported experiments on MNIST and corrupted MNIST with FCNNs and LeNet, m-Sharpness correlated with the generalization gap at least comparably to normalized sharpness, while the exact trace computation was dramatically faster than a naive Hessian-based baseline. The paper positions m-Sharpness as a scale-invariant measure that is both exact and tractable in standard softmax-classification settings (Ibayashi et al., 2021).

4. $m$6-Sharpness in SAM, mSAM, and micro-batch optimization

A second, now widespread, use of the term arises from SAM. The original SAM objective is

$m$7

with the usual first-order perturbation for $m$8,

$m$9

Foret et al. further introduced the idea that sharpness evaluated on smaller subsets can be more predictive of generalization; later work formalized this as $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$0-sharpness, with $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$1 denoting the subset or micro-batch size (Foret et al., 2020).

In mSAM, a mini-batch $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$2 is split into disjoint shards $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$3, and each shard receives its own SAM perturbation: $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$4 Under the first-order approximation,

$$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$5

This differs from standard SAM because the perturbation is nonlinear in the batch gradient; averaging shardwise SAM updates is therefore not equivalent to a single full-batch perturbation.

Two empirical studies reported that mSAM improved test performance over both SAM and vanilla training across vision and NLP workloads, while remaining substantially cheaper than a naive $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$6-fold cost model would suggest. On CIFAR-100, one study reported ResNet50 test accuracy of $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$7 for vanilla training, $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$8 for SAM, and $$\min_{\alpha}\operatorname{Tr}[\mathbf{H}_{\alpha(\theta)}]$$9 for mSAM, with corresponding largest-Hessian-eigenvalue estimates $m$0, $m$1, and $m$2; for WRN-28-10, the reported values were $m$3, $m$4, and $m$5, with $m$6, $m$7, and $m$8 (Behdin et al., 2022). A related analysis framed the sharpness ordering at the edge of stability as

$m$9

for SGD, SAM, and mSAM, respectively, and emphasized that $m$0 should be treated as a hyperparameter rather than as the number of accelerators (Behdin et al., 2023).

A 2025 SDE-based account gave a different explanation of the same phenomenon. In that formulation, the performance gain from decreasing micro-batch size was itself called m-sharpness. The main claim was that smaller micro-batches amplify an implicit variance-based regularizer tied to the stochastic-gradient-noise covariance $m$1. For m-USAM, the drift was derived as

$m$2

This makes the coefficient of $m$3 scale as $m$4, so the regularization strengthens as $m$5 decreases (Luo et al., 22 Sep 2025).

5. Theoretical revisions, critiques, and limits

The sharpness literature does not treat $m$6-Sharpness as an uncontested explanatory primitive. One line of criticism argues that SAM’s perturbed loss is not itself a faithful curvature proxy. GSAM formalizes this by introducing the surrogate gap

$m$7

and proving that at a local minimum, for small $m$8,

$m$9

GSAM then modifies SAM by adding an orthogonal ascent component,

$\theta_{2n}(p-2\theta_m(p))$0

with the stated goal of reducing the surrogate gap without changing the perturbed-loss directionally. The paper positions this as a more faithful route to low sharpness than minimizing perturbed loss alone (Zhuang et al., 2022).

Another critique is that neither PAC-Bayes-style arguments nor “flat minima” by themselves explain why low-$\theta_{2n}(p-2\theta_m(p))$1 SAM should generalize better. For diagonal linear networks, one theoretical account instead attributes the effect to implicit bias: SAM changes the effective initialization scale, and the change is stronger for smaller $\theta_{2n}(p-2\theta_m(p))$2. In that analysis, $\theta_{2n}(p-2\theta_m(p))$3-SAM and $\theta_{2n}(p-2\theta_m(p))$4-SAM select interpolating solutions minimizing the same potential $\theta_{2n}(p-2\theta_m(p))$5, but with reduced effective scales $\theta_{2n}(p-2\theta_m(p))$6 and $\theta_{2n}(p-2\theta_m(p))$7, producing a stronger sparsity bias for smaller $\theta_{2n}(p-2\theta_m(p))$8 (Andriushchenko et al., 2022).

The broader sharpness-generalization link is also contested empirically and theoretically. One paper identified three scenarios for two-layer ReLU networks: flatness can imply generalization; flattest models can fail to generalize and sharpness minimization can fail as well; and flattest models can fail to generalize even though sharpness-minimization algorithms still do generalize. In that work, sharpness was taken as $\theta_{2n}(p-2\theta_m(p))$9, with the interpolation identity

$$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$0

for mean squared error (Wen et al., 2023). A separate large-scale empirical study of adaptive, reparametrization-invariant sharpness found that sharpness often correlated more strongly with training hyperparameters such as learning rate than with generalization, and in multiple cases sharper minima had lower out-of-distribution error (Andriushchenko et al., 2023).

Taken together, these results suggest that m-Sharpness is best understood as a family of structured sharpness notions rather than as a universally sufficient explanation of generalization.

6. Other formal meanings: percolation and quantum measurement

Outside machine learning, the term appears in mathematically unrelated forms. In Bernoulli percolation on a locally finite, transitive, infinite graph, the paper on coupling-based sharpness proves the multiscale estimate

$$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$1

with $$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$2 in the theorem statement of the paper. This “$$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$3-sharpness” estimate links a parameter decrease controlled by the short-scale one-arm probability $$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$4 to an exponential-in-$$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$5 suppression factor. It implies exponential decay in the subcritical phase and the mean-field lower bound

$$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$6

in the supercritical phase (Vanneuville, 2022).

A distinct 2026 quantum-information usage defines m-Sharpness as a basis-independent measurement sharpness obtained by maximizing a POVM’s coherence over all unitary basis changes: $$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$7 The associated distance-based sharpness monotone is

$$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$8

and for POVMs with a common eigenbasis the paper proves

$$C_D^{\max}(\mathsf A)=\max_U C_D(U^\dagger \mathsf A U)$$9

The construction is faithful in the sense that

$$f(x\mid \theta) = \mathbf{W}_{D}\phi\!\left( \mathbf{W}_{D-1}\phi \cdots \phi(\mathbf{W}_{1}x)\cdots \right),$$0

so maximal coherence vanishes exactly on trivial measurements (Baek et al., 7 Jul 2026).

These non-ML usages underscore the main encyclopedic point: m-Sharpness is not a single invariant concept shared across disciplines. It is a recurrent label for constructions that refine naïve sharpness by introducing an optimization over equivalence classes, a dependence on subset scale, or a multiscale control parameter.