arxiv: 2605.11291 · v1 · submitted 2026-05-11 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Optimal Representations for Generalized Contrastive Learning with Imbalanced Datasets

Thuan Nguyen , Shuchin Aeron , D. Richard Brown III , Prakash Ishwar

Authors on Pith no claims yet

Pith reviewed 2026-05-13 02:27 UTC · model grok-4.3

classification 💻 cs.LG

keywords contrastive learningimbalanced dataneural collapseminority collapseoptimal geometryconvex optimizationrepresentation learning

0 comments

The pith

For imbalanced contrastive learning, optimal representations collapse to class means with angles set by class proportions.

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

The paper establishes that, for a broad class of contrastive losses, the optimal learned representations cause all points from the same class to coincide at their class mean. The means themselves arrange in a symmetric way whose angles are completely determined by the relative frequencies of the classes; this arrangement is recovered by solving one convex optimization problem. When the imbalance ratio grows large enough, the theory shows that all minority classes merge into one common vector, a collapse whose onset depends on the loss details and the number of negative pairs drawn per example. A sympathetic reader cares because this gives an exact, computable description of the representational geometry that appears in practice and explains why minority classes can be lost under imbalance.

Core claim

For a large, generalized family of contrastive learning losses, when the representation dimension is greater than the number of classes, the optimal representations of all samples from the same class collapse to their class means and their geometry exhibits an angular symmetry structure that is determined by the relative class proportions. In general, the geometry can be determined by solving a convex optimization problem. When class imbalance is extreme, contrastive learning exhibits minority collapse where all samples from the minority classes collapse into a single vector whenever the class imbalance exceeds a threshold depending on the loss and number of negative samples.

What carries the argument

The convex optimization problem whose solution gives the angular symmetry structure of the optimal class means under a generalized contrastive loss.

If this is right

The geometry of optimal representations is obtained by solving a convex program for any loss in the family.
Minority collapse occurs once the imbalance ratio surpasses a threshold fixed by the loss regularity and the number of negative samples.
The balanced case recovers the known equiangular tight frame geometry as a special case.
These predictions are confirmed by numerical simulations on imbalanced datasets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Training procedures could incorporate the convex program as a regularizer to control the desired geometry.
Investigating finite sample or non-converged training dynamics would reveal how close real models come to this optimal geometry.
The minority collapse threshold offers a quantitative criterion for deciding when class reweighting or oversampling is required.

Load-bearing premise

The analysis assumes the loss function satisfies certain regularity conditions, the embedding dimension exceeds the number of classes, and training reaches the global optimum of the loss.

What would settle it

Train a contrastive model to convergence on a synthetic dataset with controlled class proportions, extract the empirical class means, and verify whether their inner products match the solution of the convex optimization and whether minority means coincide exactly above the predicted imbalance threshold.

Figures

Figures reproduced from arXiv: 2605.11291 by D. Richard Brown III, Prakash Ishwar, Shuchin Aeron, Thuan Nguyen.

**Figure 2.** Figure 2: Minority collapse in heavily imbalanced datasets. The [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

read the original abstract

In this paper, we provide a computable characterization of the geometry of optimal representations in Contrastive Learning (CL) when the classes are imbalanced. When classes are balanced and the representation dimension is greater than the number of classes, it is well-known that the optimal representations exhibit Neural Collapse (NC), i.e., representations from the same class collapse to their class means and the class means form an Equiangular Tight Frame (ETF). For imbalanced classes and a large, generalized family of CL losses, we prove that the optimal representations of all samples from the same class collapse to their class means and their geometry exhibits an angular symmetry structure that is determined by the relative class proportions. In general, we show that the geometry can be determined by solving a convex optimization problem. Exploiting this symmetry structure, we analytically investigate a special case where class imbalance is extreme and prove that CL exhibits a phenomenon called Minority Collapse (MC) where all samples from the minority classes (classes with small probabilities) collapse into a single vector, whenever the class imbalance exceeds a threshold, which in turn depends on the regularity properties of the CL loss used and on the number of negative samples. Numerical results are provided to illustrate these phenomena and corroborate the theoretical results. We conclude by identifying a number of open problems.

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

This paper extends neural collapse to imbalanced contrastive learning with a convex characterization of optimal geometry and a new minority collapse phenomenon.

read the letter

The punchline is that this work characterizes the optimal geometry for generalized contrastive learning under imbalance: within-class collapse to means, proportion-determined angular symmetry via convex optimization, and a threshold for minority collapse where small classes all map to one vector. The paper does a good job extending the balanced neural collapse results to the imbalanced case that matters in practice. They prove the collapse and symmetry for a large family of losses, reduce the geometry to a convex program that depends only on class proportions, and then analytically find when minority collapse happens as a function of the imbalance level, loss regularity, and number of negatives. The numerical experiments illustrate both the symmetry and the collapse threshold, which helps make the theory concrete. Soft spots are minor. The results assume the representation dimension exceeds the number of classes, the loss meets regularity conditions, and training reaches the global optimum. These are stated clearly, but in real training one might not hit the exact global min, so the predictions are more about the loss landscape than guaranteed behavior. The convex optimization is presented as computable, but no details on its complexity or solvers are given here, though that's probably secondary. This paper is for theorists in machine learning who study contrastive learning, self-supervised methods, and representation geometry. Anyone working on long-tailed data or imbalance in SSL will find the minority collapse concept and the convex characterization useful. I would bring it to the next reading group to discuss the proof strategy and the open problems. I would cite the minority collapse result in related theoretical work. It deserves peer review because the contributions are novel, the assumptions are transparent, and the evidence from proofs and numerics is sufficient to merit referee scrutiny.

Referee Report

1 major / 3 minor

Summary. The paper claims that for a generalized family of contrastive learning (CL) losses with imbalanced class proportions, the optimal representations exhibit within-class collapse to their class means, with the geometry of these means displaying an angular symmetry structure fully determined by the relative class proportions; this geometry is characterized as the solution to a convex optimization problem. In the extreme-imbalance regime, it further proves a 'Minority Collapse' (MC) phenomenon in which all minority-class samples collapse to a single vector once imbalance exceeds a threshold that depends on loss regularity properties and the number of negative samples. The results extend the balanced-case Neural Collapse (NC) geometry and are supported by numerical experiments.

Significance. If the stated regularity conditions on the loss family, the dimension assumption (d larger than the number of classes), and global optimality are satisfied, the work supplies a computable, convex characterization of optimal CL geometries under imbalance together with an explicit, falsifiable threshold for MC. These are concrete strengths: the reduction to a convex program and the analytic derivation of the MC threshold constitute parameter-free predictions that can be checked numerically, extending the NC literature in a precise way.

major comments (1)

[§3 (symmetry characterization) and §4 (MC threshold)] The central proofs (likely §3–4) rely on the representation dimension being sufficiently large relative to the number of classes and on the optimizer attaining the global minimum of the loss; these assumptions are load-bearing for both the symmetry characterization and the MC threshold, yet the manuscript does not provide a quantitative lower bound on d or a verification that the convex program indeed corresponds to a global minimizer of the original non-convex CL objective.

minor comments (3)

[Main theoretical section] The convex optimization problem that determines the angular geometry should be stated explicitly (including the objective and constraints) in the main text rather than deferred to an appendix, so that the dependence on class proportions is immediately visible.
[Experimental section] Numerical experiments should report the precise CL loss (e.g., InfoNCE with temperature), the exact imbalance ratios, and the number of negative samples used, together with a direct comparison of the observed angles against the convex-program solution.
[Introduction / §4] The term 'Minority Collapse' is introduced as a new phenomenon; a short paragraph contrasting it with standard NC and with other collapse notions in the literature would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and for highlighting the load-bearing assumptions in our proofs. We address the major comment below.

read point-by-point responses

Referee: [§3 (symmetry characterization) and §4 (MC threshold)] The central proofs (likely §3–4) rely on the representation dimension being sufficiently large relative to the number of classes and on the optimizer attaining the global minimum of the loss; these assumptions are load-bearing for both the symmetry characterization and the MC threshold, yet the manuscript does not provide a quantitative lower bound on d or a verification that the convex program indeed corresponds to a global minimizer of the original non-convex CL objective.

Authors: We agree that these assumptions are central. The proofs establish that any global minimizer must exhibit within-class collapse to class means whose angular geometry solves the stated convex program; the dimension assumption enters to ensure the means can realize arbitrary angles in the collapsed regime. We will revise the manuscript to explicitly state the quantitative lower bound d ≥ K (where K is the number of classes) and add a brief justification in §3, noting that this is the minimal dimension allowing the class means to span the necessary (K-1)-dimensional configuration without interference, consistent with the balanced NC case. Regarding global optimality, our results characterize the geometry attained at global minimizers of the non-convex loss rather than proving from scratch that the convex program's solution is always achieved; the reduction to the convex program follows after proving collapse must occur at optimality. A complete landscape analysis confirming attainment without further assumptions lies outside the current scope, though it is consistent with the numerical experiments. We will add a short discussion remark acknowledging this and listing it as an open question. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation reduces to external class proportions and stated loss assumptions

full rationale

The central results characterize minimizers of a generalized family of contrastive losses by proving within-class collapse to means and reducing inter-class angular geometry to a convex program whose sole data-dependent inputs are the given class proportions together with explicit regularity conditions on the loss and the number of negative samples. The Minority Collapse threshold is obtained directly from the same symmetry analysis without any fitted parameters, self-referential definitions, or load-bearing self-citations. All steps are self-contained once the stated assumptions (sufficient representation dimension and global optimality) are granted; no equation or claim reduces by construction to a prior fitted quantity or to an unverified self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claims rest on assumptions about the loss family and optimization landscape; no free parameters are explicitly fitted beyond data properties like class proportions.

axioms (1)

domain assumption Contrastive loss functions belong to a generalized family possessing regularity properties that enable the collapse proofs.
This is invoked to prove the collapse to class means and the minority collapse under extreme imbalance.

invented entities (1)

Minority Collapse no independent evidence
purpose: Describes the merging of minority class representations into a single vector under severe imbalance.
Introduced as a new phenomenon specific to this analysis.

pith-pipeline@v0.9.0 · 5534 in / 1397 out tokens · 73708 ms · 2026-05-13T02:27:16.954393+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

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that the optimal representations of all samples from the same class collapse to their class means and their geometry exhibits an angular symmetry structure that is determined by the relative class proportions. In general, we show that the geometry can be determined by solving a convex optimization problem.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the lower bound is a strictly convex function of the Gram matrix whose entries are the pairwise inner products of unit-norm class mean feature vectors

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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