MeanSE: Error Metrics and Generative Flows

• MeanSE is a technical term with three distinct definitions: a limiting MSE in regression, an error metric in DP training, and a generative speech model in enhancement.
• In high-dimensional regression, it is derived via AMP, CGMT, and LOO, yielding method-independent asymptotic risk under Gaussian design assumptions.
• In DP training and speech enhancement, MeanSE captures averaged error metrics through workload query evaluation and time-averaged velocity, emphasizing task-specific surrogate quality.

MeanSE is an overloaded technical term with distinct meanings across recent arXiv literatures. In high-dimensional regression, it denotes the limiting mean squared error predicted by scalar state equations for estimators such as approximate message passing (AMP) and convex estimators analyzed through the Convex Gaussian Min-Max Theorem (CGMT). In differentially private optimization, it denotes the root mean square error averaged over a workload’s queries, typically after correlated Gaussian noise is propagated through a schedule-dependent workload matrix. In speech enhancement, MeanSE is the name of a generative model that replaces an instantaneous flow field with a mean, time-averaged velocity field to obtain strong single-step enhancement. These usages share an emphasis on low-dimensional error characterizations, but they refer to different mathematical objects and should not be conflated (Luo et al., 2022, Kalinin et al., 22 Nov 2025, Wang et al., 25 Sep 2025).

1. Terminological scope

In current usage, “MeanSE” appears in at least three technically distinct roles. First, it is an asymptotic risk quantity in high-dimensional statistics. Second, it is an error metric in matrix-factorized differentially private training. Third, it is a proper model name in generative speech enhancement. This suggests that the term is best interpreted contextually rather than as a single standardized notion.

Usage	Definition	Context
MeanSE as limiting risk	$$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big]$$	State evolution for high-dimensional regression
MeanSE as workload error	$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$	DP-SGD with correlated noise and learning-rate schedules
MeanSE as model name	Efficient generative speech enhancement with mean flows	Single-channel speech enhancement in the STFT domain

A recurring misconception is to treat these as interchangeable. They are not. The first is a limiting MSE under asymptotic Gaussian design, the second is an averaged RMSE over workload outputs, and the third is an architecture/training framework whose name happens to contain the same string.

2. MeanSE in high-dimensional regression

In high-dimensional regression, MeanSE refers to the limiting mean squared error predicted by state equations. The canonical AMP setup considers

$$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$

With AMP iteration

$$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$

the state evolution recursion is

$$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$

and the iteration-wise MSE is

$$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$

If $\tau_t \to \tau_\star$ and $\theta_t \to \theta_\star$, then

$$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$

and

$$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$

The central result is that apparently different scalar characterizations derived by AMP, CGMT, and leave-one-out (LOO) arguments are equivalent after explicit parameter transformations. For robust M-estimation, the mappings

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$0

and

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$1

make the different state-equation systems identical. For LASSO, the mapping

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$2

aligns the CGMT scalar system with the AMP state evolution. For logistic regression, the mapping

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$3

plays the same role. Under these transformations, the predicted MeanSE is method-independent in the high-dimensional Gaussian-design limit, provided the regularity assumptions hold: $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$4 with $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$5, Gaussian design, separable prior or denoiser, and uniqueness or stability of the state-evolution fixed point (Luo et al., 2022).

3. MeanSE as an error metric in private training

In differentially private training with learning-rate schedules and correlated noise, MeanSE is defined for a workload $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$6 and noise covariance $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$7 by

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$8

The corresponding worst-case metric is

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$9

Here the workload is induced by scheduled SGD updates. With $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$0 iterations, base learning rate $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$1, schedule $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$2, and actual step size $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$3, the cumulative dependence on clipped gradients is represented by

$$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$4

where $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$5 is the lower-triangular matrix of ones and $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$6. Constant, exponential, polynomial, linear, cosine, step, and warmup-plus-decay schedules all fit this formulation.

Matrix-factorization mechanisms realize correlated noise through $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$7 with $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$8, so that $$y = A x_0 + w, \quad A_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}\Big(0,\frac{1}{m}\Big),\quad w \sim \mathcal{N}(0,\sigma_w^2 I_m), \quad \delta = \frac{m}{n}.$$9. Then

$$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$0

The paper factors the workload as $$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$1, adds Gaussian noise to $$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$2, and returns $$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$3; this yields

$$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$4

after privacy calibration.

The schedule-aware contribution is a Toeplitz square-root construction. For exponential decay $$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$5, one sets

$$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$6

with explicit lower-triangular form

$$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$7

and $$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$8. For exponential decay, this achieves

$$x^{t+1} = \eta\big(x^t + A^\top z^t; \theta_t\big), \qquad z^t = y - A x^t + \frac{z^{t-1}}{\delta}\,\Big\langle \eta'\big(x^{t-1} + A^\top z^{t-1}; \theta_{t-1}\big)\Big\rangle,$$9

and

$$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$0

The schedule-aware factorization improves over prefix-sum factorizations under both MaxSE and MeanSE, with the strongest theoretical and empirical gains under stronger decay schedules (Kalinin et al., 22 Nov 2025).

4. MeanSE as a generative speech enhancement model

As a model name, MeanSE denotes an efficient generative speech enhancement system based on mean flows. The task is single-channel speech enhancement in the STFT domain, with clean speech $$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$1, noisy speech $$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$2, flow state $$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$3, and conditional velocity field $$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$4. Standard conditional flow matching learns the instantaneous field in

$$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$5

along the Gaussian conditional path

$$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$6

with

$$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$7

The corresponding conditional flow-matching loss is

$$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$8

MeanSE replaces the instantaneous target with the average velocity over an interval $$\tau_t^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big],$$9: $$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$0 It is trained by regressing this analytic mean-flow target: $$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$1 For $$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$2 intervals, inference updates

$$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$3

and the single-step specialization is

$$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$4

The architecture uses NCSN++ with complex STFT input, $$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$5, hop length $$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$6, and spectral image size $$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$7. Time conditioning uses separate Gaussian Fourier feature embeddings for $$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$8 and $$\mathrm{MSE}_t = \mathbb{E}\Big[\big(\eta(X_0 + \tau_{t-1} Z; \theta_{t-1}) - X_0\big)^2\Big].$$9, fused into a single embedding. Two stabilization strategies are central: flow field mix-up, in which $\tau_t \to \tau_\star$0 is used with a specified “flow ratio,” and time interval curriculum learning with maximum interval lengths $\tau_t \to \tau_\star$1. Training uses VoiceBank-DEMAND at $\tau_t \to \tau_\star$2 kHz, with learning rate $\tau_t \to \tau_\star$3 from scratch and $\tau_t \to \tau_\star$4 for fine-tuning, weight decay $\tau_t \to \tau_\star$5, $\tau_t \to \tau_\star$6, batch size $\tau_t \to \tau_\star$7, and approximately $\tau_t \to \tau_\star$8 epochs.

On VoiceBank-DEMAND, the 1-NFE comparison against FlowSE reports: FlowSE, NFE$\tau_t \to \tau_\star$9, SIG $\theta_t \to \theta_\star$0, BAK $\theta_t \to \theta_\star$1, OVRL $\theta_t \to \theta_\star$2, UTMOS $\theta_t \to \theta_\star$3, WVMOS $\theta_t \to \theta_\star$4, NISQA $\theta_t \to \theta_\star$5, PESQ $\theta_t \to \theta_\star$6, ESTOI $\theta_t \to \theta_\star$7; MeanSE, NFE$\theta_t \to \theta_\star$8, SIG $\theta_t \to \theta_\star$9, BAK $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$0, OVRL $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$1, UTMOS $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$2, WVMOS $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$3, NISQA $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$4, PESQ $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$5, ESTOI $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$6. On out-of-domain WHAMR!, the 1-NFE comparison gives FlowSE: DNSMOS $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$7, UTMOS $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$8, WVMOS $$\tau_\star^2 = \sigma_w^2 + \frac{1}{\delta}\,\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big],$$9, NISQA $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$0; MeanSE: DNSMOS $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$1, UTMOS $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$2, WVMOS $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$3, NISQA $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$4. The ablation on flow field mix-up further shows that ratio $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$5 is unstable and that larger ratios improve both in-domain and out-of-domain behavior, with $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$6 used in the main results (Wang et al., 25 Sep 2025).

5. Relation to MSE and adjacent error criteria

Although the term “MeanSE” contains “MSE,” its interpretation depends on context, and raw MSE relationships do not automatically transfer to other evaluation criteria. For ordinary prediction, the population MSE between predictions $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$7 and reference $$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$8 is

$$\mathrm{MeanSE}=\mathbb{E}\Big[\big(\eta(X_0 + \tau_\star Z; \theta_\star) - X_0\big)^2\Big].$$9

The concordance correlation coefficient is

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$00

Writing

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$01

one gets

$$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$02

Because $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$03 can vary for fixed MSE, the mapping is many-to-many: the same MSE can correspond to different values of $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$04, and a lower MSE does not imply a higher concordance correlation. This is directly relevant to MeanSE terminology. In the high-dimensional-regression sense, MeanSE is an asymptotic risk target. In the DP-workload sense, it is already a transformed RMSE over multiple linear queries. In speech enhancement, it is not itself the evaluation metric, since reported performance also uses PESQ, ESTOI, DNSMOS P.835, WVMOS, UTMOS, and NISQA. A plausible implication is that “MeanSE” should be read as a task-specific surrogate or summary quantity rather than as a universal proxy for perceptual or agreement-oriented quality (Pandit et al., 2019).

6. Assumptions, limitations, and interpretive issues

Each usage of MeanSE is coupled to a specific asymptotic or algorithmic regime. In high-dimensional regression, the equivalence of AMP-, CGMT-, and LOO-derived state equations relies on high-dimensional asymptotics, Gaussian design, separable priors or denoisers, and regularity conditions such as convexity, differentiability, and fixed-point uniqueness. In private training, the schedule-aware MeanSE theory depends on workload factorization, DP calibration through $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$05 or $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$06, and regularity conditions on adjacent schedule differences; the multi-epoch theory further requires $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$07-min separation and banded inverse constructions. In speech enhancement, the reported MeanSE model depends on the Gaussian conditional path, analytic average-velocity targets, FlowSE initialization, flow field mix-up, and curriculum learning, and purely average-velocity training with ratio $$\mathrm{MeanSE}(W,\Sigma)=\sqrt{(1/m)\operatorname{tr}(W\Sigma W^\top)}$$08 is reported as unstable. These constraints imply that cross-domain translation of the term is usually semantic rather than mathematical: the shared label identifies an averaging principle over error or velocity, but not a single invariant object (Luo et al., 2022, Kalinin et al., 22 Nov 2025, Wang et al., 25 Sep 2025).

A second interpretive issue concerns notation. In the regression literature, MeanSE is literally a limiting mean squared error. In the privacy literature, MeanSE includes a square root and is therefore an averaged RMSE. In the speech-enhancement literature, MeanSE is a model name rather than an error formula. The technical literature therefore uses the same string for a risk limit, an evaluation metric, and a generative architecture. Careful reading of definitions is essential whenever the term appears without qualification.