Does a Shared Temperature Imply a Shared Angular Scale in Probabilistic Contrastive Learning?

Ningkang Peng; Qianfeng Yu; Jingyang Mao; Xiaoqian Peng; Tingyu Lu; Peirong Ma; Yanhui Gu

cs.LGSep 30, 2026

Does a Shared Temperature Imply a Shared Angular Scale in Probabilistic Contrastive Learning?

Authors: Ningkang Peng, Qianfeng Yu, Jingyang Mao, Xiaoqian Peng, Tingyu Lu, Peirong Ma, Yanhui Gu

Organizations: Nanjing Normal University · Nanjing University of Chinese Medicine · Tohoku University

Abstract

In probabilistic contrastive learning, a shared temperature is commonly interpreted as a shared similarity scale, but this interpretation does not hold for high-dimensional distributional class representations. We study the exact von Mises-Fisher (vMF) probabilistic score used by ProCo when representation dimension and class concentration grow jointly. We prove that the score retains a class-dependent leading angular gain $g_c=A_c/τ$ , where $A_c$ is the mean resultant length. This gain enters Softmax competition, pairwise decision boundaries, and feature gradients. On real CIFAR-LT, ImageNet-LT, and iNaturalist representations, the theory accurately predicts boundary movements and local gradient changes under the full vMF score. Classwise temperature adjustment also changes the cosine-zero intercept and finite-dimensional response. We construct intercept-preserving and Pure Angular controls to separate the leading gain from these accompanying changes. Complete gain equalization yields a shared-scale cosine prototype rule at leading order; a finite-dimensional margin condition guarantees agreement of the two classifiers. Across 16 frozen representation settings, prediction agreement is 98.43-99.99%, with disagreements concentrated at small cosine margins. In controlled contrastive-only training with the training-frequency prior, Pure Angular editing improves both learned representations at all tested CIFAR-10/100 imbalance factors and retains positive changes on ImageNet-LT. Thus vMF concentration not only describes class distributions, but also forms a decision and learning scale in high-dimensional probabilistic contrastive learning.

Figures & tables

Dataset / IF	Feature	ProCo	Direct temperature $\beta=0$	Intercept- preserving $\beta=0$	Pure Angular $\beta=0$	$\Delta$
CIFAR-10 / 10	Encoder	$86.28\pm 0.09$	$86.52\pm 0.15$	$86.54\pm 0.11$	$86.93\pm 0.10$	$+0.65$
	Projection	$87.10\pm 0.06$	$87.55\pm 0.08$	$87.65\pm 0.16$	$87.96\pm 0.13$	$+0.86$
CIFAR-10 / 50	Encoder	$75.73\pm 0.12$	$75.93\pm 0.19$	$76.02\pm 0.10$	$76.14\pm 0.17$	$+0.41$
	Projection	$77.66\pm 0.07$	$77.64\pm 0.11$	$78.46\pm 0.14$	$78.83\pm 0.09$	$+1.17$
CIFAR-10 / 100	Encoder	$68.24\pm 0.16$	$67.89\pm 0.13$	$68.13\pm 0.13$	$68.43\pm 0.11$	$+0.19$
	Projection	$72.65\pm 0.10$	$72.48\pm 0.14$	$72.74\pm 0.08$	$73.12\pm 0.16$	$+0.47$

Table 1: Representation accuracy after contrastive-only training. Long-tail linear-probe accuracy (%) after feature learning with the training-frequency prior. All interventions use

\beta=0

. Results report mean

\pm

sample standard deviation over three seeds.

\Delta

is the difference between the displayed Pure Angular and ProCo means in percentage points; darker green marks a larger increase. Readout protocols are given in Section 3.4 and Appendices I.2 and N .

Figure 1: Class-dependent angular gain at a shared temperature. (a) A training-selected CIFAR-100-LT pair in a boundary-sign-preserving spherical projection. Solid and dashed curves show original and equalized linear-mean boundaries at zero prior; shading shows the original regions. (b) Exact centered scores and linear-mean approximations for the same pair. (c) All-class empirical distributions of $A_{c}/\tau_{0}$ in three projection spaces, with equal weight per class.

Figure 2: Accuracy and residual orders of the joint expansion. Left: median log error ratio of second-order joint and fixed-dimensional approximations over the prespecified temperature–cosine grid. Right: observed residual orders for successive approximations; diamonds mark theoretical orders, boxes show interquartile ranges, and whiskers use $1.5$ interquartile ranges. Formulas, validity domains, and fitting details are in Theorem 4 and Appendix J .

Figure 3: Predicted and exact boundary displacement. Left: exact and joint-leading margins near the roots of a fixed CIFAR-100-LT pair. Right: theory versus exact displacement for all preselected pairs with a unique root in both conditions at zero prior. The legend gives comparable-pair coverage. The CIFAR-10 exhaustive extension has MAE

0.0116^{\circ}

(45/45 pairs); for CIFAR-100, ImageNet-LT, and iNaturalist, displacement MAEs are

0.119^{\circ}

0.0027^{\circ}

, and

0.0042^{\circ}

, respectively.

Figure 4: Intercept drift under matched gain interventions. (a) Maximum absolute classwise change in the cosine-zero score (linear scale below 0.01, logarithmic above). Open circles denote four separately fixed original ProCo states; stars denote interventions on the independently trained IF10 uniform-prior temperature-failure state. (b) Exact cross-entropy on the same 500 queries from the failure state. Within each state, queries, statistics, and prior are fixed; the three interventions share

g_{c}^{*}=\bar{g}

Dataset / IF	Feature	Original vMF	Cosine	Pure Angular	Agreement
CIFAR-10 / 10	Encoder	85.21	85.82	85.90	99.74
	Projection	87.85	87.84	87.83	99.99
CIFAR-10 / 50	Encoder	81.47	80.76	80.72	99.46
	Projection	80.72	80.58	80.58	99.95
CIFAR-10 / 100	Encoder	78.76	78.19	78.02	99.51
	Projection	77.81	77.59	77.58	99.93

Table 2: Frozen classification after complete gain equalization. Accuracy and cosine–Pure Angular prediction agreement (%) on identical features and class statistics, with zero evaluation offset and

\beta=0

. CIFAR and ImageNet use original contrastive-only models; iNaturalist uses the official ProCo model. Full realization comparisons are in Appendix G ; evaluation protocols are given in Appendices I and J .

Appendix figures & tables30 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Geometric effects of score parameters. Temperature, bias, concentration, and prototype angle affect pairwise and one-versus-rest boundaries; class count affects only the latter. Joint B0/B2 are truncations through orders zero/two.

Figure 6: Finite-dimensional approach to the tail-error transition. (a,b) The transition narrows around $R_{t}=0$ . (c) Exact one-dimensional errors approach the large-deviation exponents. (d) Exact ProCo and the leading rule agree. Zero Monte Carlo counts are displayed at half a count and are not treated as zero probability.

Figure 7: Reversing the prior ordering can reverse the head–tail exponent order; the prior condition in the theorem is substantive.

Dataset / IF	Feature	Original	$q^{\rm temp}$	$q^{\rm preserve}$	$q^{\rm angular}$	$\Delta$
CIFAR-10 / 10	Encoder	85.21	85.92	85.84	85.90	$+0.69$
	Projection	87.85	87.83	87.83	87.83	$-0.02$
CIFAR-10 / 50	Encoder	81.47	80.71	80.72	80.72	$-0.75$
	Projection	80.72	80.58	80.58	80.58	$-0.14$
CIFAR-10 / 100	Encoder	78.76	78.00	78.12	78.02	$-0.74$
	Projection	77.81	77.58	77.59	77.58	$-0.23$

Appendix

Table 3: Frozen accuracy across score realizations. Accuracy (%) at

\beta=0

and zero prior offset.

\Delta

is Pure Angular minus original vMF in percentage points; darker green indicates a larger increase.

Dataset / IF	Feature	Disagree	Low-margin	Certified / Total
CIFAR-10 / 10	Encoder	26	26	9,763 / 10,000
	Projection	1	1	9,982 / 10,000
CIFAR-10 / 50	Encoder	54	54	9,717 / 10,000
	Projection	5	5	9,969 / 10,000
CIFAR-10 / 100	Encoder	49	49	9,655 / 10,000
	Projection	7	7	9,960 / 10,000

Appendix

Table 4: Prediction disagreements and margin certificates. “Low-margin” reports the number of disagreements falling in the lowest cosine-gap decile. Certified queries satisfy

\bar{g}m_{\cos}>D

; all certified queries agree.

Dataset / IF	Feature	Mean $\|\Delta L\|$	Grad. error (%)	Median grad. cosine
CIFAR-100 / 10	Encoder	0.01141	1.555	0.99999712
	Projection	0.01751	1.624	0.99996033
CIFAR-100 / 50	Encoder	0.00917	1.524	0.99999679
	Projection	0.02206	1.592	0.99996161
CIFAR-100 / 100	Encoder	0.00832	1.448	0.99999724
	Projection	0.02634	1.595	0.99995594

Appendix

Table 5: Loss and gradient differences between Pure Angular and cosine. Complete equalization with zero prior offset. Gradient error is

100\|G_{\rm angular}-G_{\cos}\|_{F}/\|G_{\rm angular}\|_{F}

, with query tangent gradients as rows of

G

. The last column reports their median cosine similarity.

Figure 8: Variance coupling in the multiclass expansion. Retaining the second-order variance term recovers the predicted third-order residual decay.

Figure 9: Support size and self-inclusion effects. (a) Resultant-length estimation error; bars show between-class SD. (b) Current-query inclusion effects; bars show average within-class query SD.

Figure 10: Class geometry and vMF goodness of fit. Left: class frequency and fitted concentration. Right: held-out projection goodness of fit.

Figure 11: Margins and fitted-score errors. Prototype-centre and random-typical margins versus reconstructed pairwise errors.

$\beta$	$\gamma$	ResNet-50	ResNeXt-50
0	-1	-1.665	-1.465
0	0	+0.195	+0.315
0	1	+0.175	-0.175
0.5	-1	-0.635	-0.610
0.5	0	+0.260	+0.240
0.5	1	+0.055	-0.040

Appendix

Table 6: Frozen gain interventions across backbones. ImageNet-LT projection-score accuracy changes in percentage points relative to ProCo (

\beta=1

) at the same prior.

Figure 12: Prior-dependent effects of angular gain. Frozen vMF accuracy changes relative to ProCo at the same prior strength

\gamma

Figure 13: Angular gain and local gradients. (a) Gain dispersion during training. (b) Relative error of leading gradient increments. Colors identify interventions; line styles identify training states. (c) Exact (solid) and leading (dashed) increments projected onto a common plane.

	CIFAR-10-LT		CIFAR-100-LT
Policy	IF50	IF100	IF50	IF100
ProCo	87.57	85.39	55.78	51.34
Reduced gain disparity	87.72	85.10	56.62	51.72
Increased gain disparity	87.56	85.14	55.99	52.05

Appendix

Table 7: Historical power-temperature intervention. Original classification-head test accuracy (%).

Figure 14: Changes in common linear readouts. Accuracy differences from ProCo at epoch 200 within each feature layer on CIFAR-100-LT IF100. Each classifier is fitted independently under the same fixed protocol. These probes are distinct from the original classification heads in Table 7 .

		Contrastive-only		Joint
Realization	$\beta$	E	P	E	P	FC
ProCo	1	92.15	92.91	92.95	93.04	92.81
Temperature	0	92.09	93.13	92.77	92.60	92.54
Temperature	0.5	92.32	92.95	92.40	92.32	92.38
Intercept-preserving	0	92.46	93.01	93.01	93.04	92.85
Intercept-preserving	0.5	92.59	93.19	92.84	92.89	92.55

Appendix

Table 8: CIFAR-10, IF1, uniform prior. Accuracy (%).

		Contrastive-only		Joint
Realization	$\beta$	E	P	E	P	FC
ProCo	1	88.97	88.63	88.57	88.28	86.32
Temperature	0	88.34	87.58	88.66	88.01	85.96
Temperature	0.5	88.29	88.21	88.20	87.84	85.96
Intercept-preserving	0	88.61	88.20	88.36	87.96	85.62
Intercept-preserving	0.5	88.51	88.24	88.34	87.96	85.83

Appendix

Table 9: CIFAR-10, IF10, uniform prior. Accuracy (%).

		Contrastive-only		Joint
Realization	$\beta$	E	P	E	P	FC
ProCo	1	88.60	88.41	88.44	88.19	87.46
Temperature	0	88.60	88.94	88.23	88.00	87.29
Temperature	0.5	88.55	88.34	88.31	88.10	87.34
Intercept-preserving	0	88.74	89.02	88.23	87.96	87.41
Intercept-preserving	0.5	88.35	88.53	88.44	87.76	87.27

Appendix

Table 10: CIFAR-10, IF10, training-frequency prior. Accuracy (%).

		Contrastive-only		Joint
Realization	$\beta$	E	P	E	P	FC
ProCo	1	82.70	81.52	83.70	82.29	76.14
Temperature	0	76.82	68.89	83.63	81.89	75.81
Temperature	0.5	80.98	78.34	83.97	82.07	76.04
Intercept-preserving	0	78.18	72.34	83.78	81.73	75.23
Intercept-preserving	0.5	81.54	79.21	83.36	81.88	76.28

Appendix

Table 11: CIFAR-10, IF50, uniform prior. Accuracy (%).

		Contrastive-only		Joint
Realization	$\beta$	E	P	E	P	FC
ProCo	1	84.86	83.91	83.42	82.04	80.52
Temperature	0	84.57	83.61	83.36	82.95	80.46
Temperature	0.5	85.16	84.31	84.25	83.37	81.24
Intercept-preserving	0	84.60	84.19	84.13	83.07	80.81
Intercept-preserving	0.5	84.38	84.04	84.60	82.93	80.79

Appendix

Table 12: CIFAR-10, IF50, training-frequency prior. Accuracy (%).

		Contrastive-only		Joint
Realization	$\beta$	E	P	E	P	FC
ProCo	1	78.28	74.65	81.64	79.55	71.25
Temperature	0	70.46	57.88	81.10	76.62	71.58
Temperature	0.5	74.56	67.21	81.93	79.16	71.28
Intercept-preserving	0	70.65	58.17	81.44	79.14	71.44
Intercept-preserving	0.5	74.82	67.63	81.42	78.43	71.31

Appendix

Table 13: CIFAR-10, IF100, uniform prior. Accuracy (%).

		Contrastive-only		Joint
Realization	$\beta$	E	P	E	P	FC
ProCo	1	82.49	81.32	81.50	80.13	77.79
Temperature	0	82.43	80.56	81.56	79.72	77.39
Temperature	0.5	82.15	80.41	81.29	79.52	77.11
Intercept-preserving	0	82.64	81.49	80.90	79.31	76.71
Intercept-preserving	0.5	82.71	81.53	81.12	78.91	76.36

Appendix

Table 14: CIFAR-10, IF100, training-frequency prior. Accuracy (%).

Figure 15: Contrastive-only training across sampling and prior settings. Balanced-support probe accuracy changes from ProCo in percentage points; colors are centered at zero.

Figure 16: Joint training with a uniform prior. Accuracy changes from ProCo in percentage points. Encoder/Projection use balanced-support probes; FC is the jointly trained head.

Figure 17: Joint training with the training-frequency prior. Accuracy changes from ProCo in percentage points. Encoder/Projection use balanced-support probes; FC is the jointly trained head.

Figure 18: Score and gradient trajectories during training. IF50 with a uniform prior: cosine-zero intercept drift and leading-gradient approximation error.

Dataset / IF	ProCo	Direct Temp.	Preserve	Pure Angular	$\Delta$
CIFAR-10 / 10	87.46	87.29	87.41	87.24	-0.22
CIFAR-10 / 50	80.52	80.46	80.81	80.79	+0.27
CIFAR-10 / 100	77.79	77.39	76.71	76.50	-1.29
CIFAR-100 / 10	58.26	58.24	57.55	58.75	+0.49
CIFAR-100 / 50	47.63	46.73	48.82	48.10	+0.47
CIFAR-100 / 100	44.67	43.15	43.54	43.88	-0.79

Appendix

Table 15: Joint-training FC-head accuracy (%) with the training-frequency prior. All modified policies use

\beta=0

\Delta

is Pure Angular minus ProCo.

\pi_{c}=\frac{n_{c}}{\sum_{j}n_{j}}.

Appendix

Algorithm 1 Controlled score realization and representation learning

Setting	CIFAR-10/100-LT	ImageNet-LT
Encoder	ResNet-32	ResNet-50
Encoder dimension	64	2048
Projection dimension	128	1024
Training epochs	200	90
Global batch size	256	256
Optimizer	SGD	SGD

Appendix

Table 16: Main feature-learning hyperparameters.

Setting	Value or protocol
Linear-classifier epochs	100; fixed final-epoch representation
Linear-classifier optimizer	SGD; batch size 256; learning rate 0.1; momentum 0.9; zero weight decay; recorded cosine schedule
Linear-classifier objective	Ordinary cross-entropy; no extra prior offset; no vMF correction
Main linear fitting pool	Corresponding long-tail training subset; ImageNet uses 115,846 fitting images
Main Encoder preprocessing	CIFAR: raw Encoder features; ImageNet: unit-normalized Encoder features in the main comparison
Projection preprocessing	Original unit-normalized Projection features

Appendix

Table 17: Frozen and linear-evaluation settings.

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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.

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