Coordinate-Wise Variance Collapse

• Coordinate-wise variance collapse is a phenomenon where certain coordinates exhibit abnormally low variability, indicating local memorization or direction-specific concentration in high-dimensional spaces.
• Baseline subtraction methods in diffusion models differentiate intrinsic data constraints from overfitting by comparing coordinate-level curvature between conditioned and unconditional settings.
• Similar collapse behaviors are observed in contrastive learning and stochastic optimization, underscoring its broader relevance for variance reduction techniques in machine learning.

Searching arXiv for directly relevant papers and adjacent work on coordinate-wise variance phenomena. Coordinate-wise variance collapse denotes a regime in which variability is suppressed not merely globally but at specific coordinates, so that selected coordinates become nearly fixed while others may remain variable. In the diffusion-model setting, it has been proposed as a geometric characterization of local memorization: memorized regions are those coordinates where local variance collapses abnormally, appearing as elevated diagonal curvature of the conditional log-density (Kim et al., 26 May 2026). More broadly, adjacent literatures use closely related notions—bounded variance, asymptotic variance zero, rank-1 or low-rank concentration, and coordinate-wise variance reduction—without always using the same term. Across these settings, the common structural idea is that variance collapse is a directional or coordinate-local degeneracy rather than a purely scalar reduction in total variance.

1. Conceptual definition and scope

The most explicit use of the term appears in work on diffusion-model memorization, which characterizes local memorization as a coordinate-wise collapse of variance (Kim et al., 26 May 2026). In that formulation, the issue is not merely that a sample has low intrinsic dimensionality, but that specific coordinates—pixels, latent spatial locations, or channels aggregated over spatial sites—have abnormally low variability and are effectively pinned to narrow values. The resulting geometry is localized rather than global: the same total degrees of freedom can be distributed across an entire image or concentrated in a small patch, and only the latter corresponds to local memorization in the sense of verbatim or template-like reproduction (Kim et al., 26 May 2026).

Related papers instantiate the same structural motif with different terminology. In Markov-source asymptotics, a single output coordinate exhibits bounded variance when its asymptotic variance coefficient is zero, yielding what that literature calls bounded variance or quasi-deterministic behavior rather than variance collapse (Kropf, 2015). In contrastive learning, fixed pairwise weights can drive representations to rank-1 solutions in deep linear networks, concentrating variance into a single direction; this is not called coordinate-wise variance collapse, but it is the closest basis-invariant analogue in that setting (Tian, 2022). In stochastic optimization and Langevin sampling, the relevant object is often the coordinate-wise second moment or mean-squared estimator error rather than the variance of a state variable, but the same collapse-versus-persistence distinction appears in the analysis of coordinate-selective updates (Ding et al., 2020, Shestakov et al., 6 Nov 2025).

This suggests a useful distinction between three nearby notions. First, there is coordinate-wise collapse of the underlying sample distribution, where selected coordinates have low conditional variance. Second, there is collapse of representation variance into a low-dimensional subspace, as in rank-1 contrastive solutions. Third, there is collapse of estimator error in coordinate-based optimization or sampling algorithms, where the sum of coordinate-wise squared errors contracts even if the underlying target distribution remains full-dimensional. These are mathematically distinct, but they share the same structural hallmark: variance becomes anisotropic and increasingly concentrated.

2. Geometric formulation in diffusion models

In diffusion models, the core objects are the unconditional and conditional score functions

$$s_\theta(x_t) \approx \nabla_{x_t}\log p(x_t), \qquad s_\theta(x_t,c) \approx \nabla_{x_t}\log p(x_t\mid c),$$

and their Hessians

$$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$

The coordinate-wise curvature signal is the diagonal of the negative Hessian,

$\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$

which is used as a coordinate-level indicator of collapse (Kim et al., 26 May 2026).

The geometric intuition is Gaussian. If $p(x)$ is Gaussian with covariance $\Sigma$, then

$-\nabla_x^2 \log p(x) = \Sigma^{-1}.$

Thus coordinates with small variance correspond to large curvature. The diffusion-model paper uses this as the starting point for a more general argument and, in its synthetic figure caption, writes

$$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$

for a linear Gaussian model, showing that two distributions with the same intrinsic dimensionality can nevertheless have very different coordinate-wise curvature patterns (Kim et al., 26 May 2026).

The formal bridge from curvature to variance is Proposition 1 of that work. With

$$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$

the paper proves

$$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$

Since $\nabla^2 \log p(x_t)$ is typically negative semidefinite near a mode, large values of $$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$0 correspond to lower posterior variance along coordinate $$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$1 (Kim et al., 26 May 2026). This gives an exact covariance–curvature identity in the diffusion setting and turns coordinate-wise curvature into a variance-collapse diagnostic.

The same paper derives the identity via Tweedie’s formula: $$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$2 which yields

$$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$3

and then

$$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$4

Combined with

$$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$5

this recovers the covariance formula above (Kim et al., 26 May 2026). In this framework, coordinate-wise variance collapse is therefore not a heuristic label but a local geometric statement about the Hessian of the log-density.

3. Baseline subtraction and the distinction between memorization and intrinsic constraint

A central complication is that coordinate-wise variance collapse alone is not sufficient evidence of overfitting. The diffusion-model paper emphasizes that some coordinates naturally have low variance because of the true data distribution or the conditioning semantics, such as prompts enforcing a black background (Kim et al., 26 May 2026). High curvature can therefore arise from intrinsic data constraints rather than memorization.

To separate these cases, the paper proposes curvature-difference methods that subtract the curvature of an underfitted baseline. With the unconditional model as baseline, the coordinate-wise metric is

$$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$6

With a less-trained checkpoint $$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$7 as baseline, the corresponding metric is

$$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$8

The intended interpretation is that intrinsic low-variance directions should already be present in an underfitted model that captures the data manifold, while excess curvature in the final model is more plausibly attributable to overfitting-driven memorization (Kim et al., 26 May 2026).

The per-coordinate quantity is then aggregated spatially in latent space. The paper states that, in practice, it works in the latent image tensor and sums across channels to obtain a spatial map,

$$H_\theta(x_t) := \nabla_{x_t} s_\theta(x_t), \qquad H_\theta(x_t,c) := \nabla_{x_t} s_\theta(x_t,c).$$9

where $\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$0 is the coordinate-wise curvature-difference tensor (Kim et al., 26 May 2026). This turns the abstract notion of coordinate-wise variance collapse into a localization method for memorized regions.

The paper further derives a score-difference proxy using the Fisher-information identity. Starting from

$\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$1

and using

$\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$2

it obtains the diagonal relation

$\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$3

This motivates the efficient proxy

$\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$4

and, for a less-trained baseline,

$\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$5

(Kim et al., 26 May 2026). The paper argues that these squared score differences approximate coordinate-wise curvature gaps, especially at late timesteps.

A plausible implication is that coordinate-wise variance collapse becomes operationally meaningful only after controlling for baseline geometry. Without that subtraction, high curvature can reflect either memorization or legitimate concentration induced by the data manifold.

4. Formal analogues in stochastic processes, optimization, and representation learning

Outside diffusion models, the same mathematical structure appears under different names.

In Markov sources, the relevant theorem concerns a single output function $\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$6 on the transitions of a finite, finally connected and finally aperiodic Markov chain. The paper shows that

$\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$7

and that bounded variance, equivalently $\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$8, occurs if and only if there exists a constant $\mathrm{diag}\!\bigl(-H_\theta(x_t,c)\bigr),$9 such that

$p(x)$0

for every directed cycle $p(x)$1 of the final component (Kropf, 2015). In the strongly connected case, this is equivalent to quasi-deterministic behavior,

$p(x)$2

Although that paper does not use the term coordinate-wise variance collapse, its bounded-variance criterion is exactly a one-coordinate collapse condition. In the multivariate case, the asymptotic covariance matrix $p(x)$3 is singular if and only if there is a nontrivial cyclewise linear relation

$p(x)$4

for all cycles $p(x)$5 (Kropf, 2015). This establishes a broader notion of directional collapse: not only coordinates, but arbitrary linear combinations can become asymptotically nonfluctuating.

In contrastive learning, the closest analogue is rank collapse. For fixed pairwise importance $p(x)$6, the max-player in deep linear contrastive learning solves

$p(x)$7

with $p(x)$8, so that representation learning reparameterizes PCA on a contrastive covariance matrix (Tian, 2022). Under the stated normalization assumptions and $p(x)$9, almost all local maxima are globally optimal aligned-rank-1 solutions,

$\Sigma$0

with objective value

$\Sigma$1

(Tian, 2022). Since the end-to-end map is then rank-1, all representation variance lies in a one-dimensional subspace. In an eigenbasis aligned with that subspace, one coordinate contains all the variance and the remaining coordinates have zero variance. The paper does not call this coordinate-wise variance collapse, but it is a direct low-rank concentration result.

In coordinate-based Langevin sampling, the issue appears as persistence or suppression of estimator variance. Naive random coordinate descent uses the one-coordinate estimator

$\Sigma$2

which is unbiased in expectation, yet the paper shows a high-variance obstruction. In a Gaussian counterexample, the per-coordinate conditional second moment satisfies

$\Sigma$3

where $\Sigma$4 (Ding et al., 2020). The point is not collapse but the failure of collapse: coordinate-wise randomization induces variance that scales linearly with dimension. The proposed RCAD estimator replaces fresh one-coordinate estimation with a memory-based scheme,

$\Sigma$5

$\Sigma$6

and satisfies

$\Sigma$7

(Ding et al., 2020). The memory mismatch contracts recursively,

$\Sigma$8

so the coordinate-wise variance obstruction is controlled indirectly through decay of stale-gradient error (Ding et al., 2020).

A closely related aggregate formulation appears in adaptive variance reduction for coordinate methods. The unified recursion controls

$\Sigma$9

and since

$-\nabla_x^2 \log p(x) = \Sigma^{-1}.$0

collapse of the aggregate mean-squared estimator error implies collapse of the sum of coordinate-wise contributions (Shestakov et al., 6 Nov 2025). For JAGUAR, a coordinate-biased method, the paper proves

$-\nabla_x^2 \log p(x) = \Sigma^{-1}.$1

while for SEGA the memory error

$-\nabla_x^2 \log p(x) = \Sigma^{-1}.$2

contracts at rate $-\nabla_x^2 \log p(x) = \Sigma^{-1}.$3 up to movement terms (Shestakov et al., 6 Nov 2025). Under the PL condition, the Lyapunov

$-\nabla_x^2 \log p(x) = \Sigma^{-1}.$4

decays linearly,

$-\nabla_x^2 \log p(x) = \Sigma^{-1}.$5

forcing the aggregate estimator error to vanish geometrically (Shestakov et al., 6 Nov 2025). This is not collapse of a model coordinate, but of coordinate-wise estimation error.

5. Methodological estimation and algorithmic realization

The diffusion-model formulation is computationally demanding because Hessian diagonals are intractable in full. The paper therefore estimates diagonal curvature differences with the Hutchinson trick: $-\nabla_x^2 \log p(x) = \Sigma^{-1}.$6 For

$-\nabla_x^2 \log p(x) = \Sigma^{-1}.$7

the diagonal estimator is

$-\nabla_x^2 \log p(x) = \Sigma^{-1}.$8

and analogously for the less-trained baseline (Kim et al., 26 May 2026). Only Hessian-vector products are needed, which makes the approach computationally feasible even though it is heavier than score differences.

The practical localization pipeline in that paper is explicit. It samples a latent state $-\nabla_x^2 \log p(x) = \Sigma^{-1}.$9 during generation, computes a score difference against either the unconditional or less-trained baseline, forms either a squared score-difference tensor or a Hutchinson-based curvature-difference tensor, sums across channels, and returns the resulting spatial map $$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$0 (Kim et al., 26 May 2026). For evaluation against masks, the maps are resized to $$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$1, globally normalized to $$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$2 independently for each metric over all evaluation samples, and thresholded by sweeping a uniform threshold across $$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$3 (Kim et al., 26 May 2026). For visualization, values are clipped at the 99th percentile; for $$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$4, negative values are clipped to zero; and for $$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$5, a $$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$6 mean filter is reported to suppress sparse outlier pixels and improve IoU (Kim et al., 26 May 2026).

Algorithmically, adjacent literatures realize coordinate-wise collapse through memory refresh or coordinate-specific updates. In RCAD-LMC, only one coordinate of the stored finite-difference table is refreshed per iteration, and the control-variate identity

$$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$7

reduces variance by preserving past coordinate information (Ding et al., 2020). In SEGA and JAGUAR, only sampled coordinates are refreshed, while unsampled coordinates remain stale. For JAGUAR,

$$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$8

so sampled coordinates collapse immediately to zero estimator error, while the aggregate error contracts at a rate determined by the sampling frequency $$-\mathrm{diag}(\nabla_x^2\log p(x)) = \mathrm{diag}\bigl((AA^\top+\sigma^2I)^{-1}\bigr)$$9 (Shestakov et al., 6 Nov 2025). This suggests a general methodological pattern: coordinate-wise collapse is often implemented not by direct diagonal variance penalization, but by selectively refreshing or contrasting coordinates against a baseline state.

6. Empirical behavior, interpretations, and limitations

The most direct empirical evidence comes from the diffusion-model memorization study. There, raw curvature

$$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$0

often highlights both memorized and intrinsically simple non-memorized regions, confirming that coordinate-wise variance collapse alone produces false positives (Kim et al., 26 May 2026). Difference-based maps

$$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$1

are more selective. On Stable Diffusion v1.4, template-verbatim only, Table 1 reports IoU $$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$2 for raw curvature, $$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$3 for $$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$4, and $$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$5 for $$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$6; on SD v1.4, all examples, it reports $$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$7 for the prior attention-based BE baseline and $$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$8 for $$p(x_t \mid x_0) = \mathcal N\!\left(\alpha_t x_0,\sigma_t^2 I\right), \qquad p(x_t)=\int p(x_t\mid x_0)\,p(x_0)\,dx_0,$$9; on SD v2.1, template-verbatim only, it reports $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$0 for BE, $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$1 for $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$2, and $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$3 for $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$4 (Kim et al., 26 May 2026). When localization maps are spatially averaged into scalar detection scores, the paper reports AUC $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$5 for both $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$6 and $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$7 on SD v1.4, and AUC $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$8 and $$\mathrm{Cov}[x_0\mid x_t] = \frac{1}{\alpha_t^2} \left( \sigma_t^4 \nabla_{x_t}^2\log p(x_t) + \sigma_t^2 I \right).$$9, respectively, on SD v2.1 (Kim et al., 26 May 2026). These results support the claim that baseline-subtracted coordinate-wise collapse is informative both locally and globally.

The paper also reports that late denoising steps work much better than early noisy steps, that even $\nabla^2 \log p(x_t)$0 Hutchinson sample is competitive, and that both unconditional and less-trained baselines are effective, including SD v1.2 as a relatively more trained baseline (Kim et al., 26 May 2026). A plausible implication is that coordinate-wise variance collapse is easiest to observe when diffusion noise is low enough that model-induced curvature is not hidden by the noise floor.

The optimization and sampling papers present a contrasting empirical message. In RCAD-LMC, naive coordinate randomization is harmful because its high variance destroys the hoped-for computational gain, while RCAD restores favorable total directional-derivative complexity through recursive variance control (Ding et al., 2020). In unified adaptive variance reduction, experiments are consistent with improved convergence of coordinate methods under adaptive stepsizes, but the paper does not report per-coordinate variance, diagonal covariance, or coordinate-wise mean-squared error trajectories (Shestakov et al., 6 Nov 2025). Likewise, the contrastive-learning paper reports competitive or improved performance of new $\nabla^2 \log p(x_t)$1-CL losses on CIFAR-10, STL-10, and CIFAR-100, but it does not measure covariance spectra or per-coordinate variances directly (Tian, 2022). These omissions matter: several adjacent literatures give rigorous structural results closely related to coordinate-wise variance collapse, yet still leave the coordinate marginals empirically unobserved.

The main misconceptions addressed by this body of work are therefore twofold. First, low variance in a coordinate is not by itself evidence of pathological memorization, because intrinsic data constraints can induce the same geometry (Kim et al., 26 May 2026). Second, global low-dimensionality does not imply local coordinate-wise collapse; a system may have few degrees of freedom distributed across the entire object rather than concentrated in a specific region (Kim et al., 26 May 2026). Conversely, a singular covariance matrix or rank-1 representation indicates directional collapse, but not necessarily collapse in the original coordinate basis (Tian, 2022, Kropf, 2015).

The principal limitation across the literature is that the strongest results are usually aggregate or basis-invariant rather than explicitly diagonal. Diffusion memorization work is exceptional in making the diagonal curvature itself the object of study (Kim et al., 26 May 2026). Markov-source theory gives exact cycle criteria for bounded-variance coordinates but does not localize collapse spatially (Kropf, 2015). Coordinate-variance-reduction papers control sums of coordinate-wise errors, not individual coordinate recursions in full generality (Ding et al., 2020, Shestakov et al., 6 Nov 2025). This suggests that coordinate-wise variance collapse is presently best understood as a family of related degeneracy phenomena rather than a single standardized object.

A plausible synthesis is that the term now designates a specific diagonalized viewpoint on concentration: variance collapse becomes meaningful when one asks not only whether variability is low, but where it is low, relative to what baseline, and whether the low-variance coordinates reflect data geometry, learned overfitting, or algorithmic error suppression.