Hyperspherical Simplex Representations for Backward-Compatible Retrieval

Simone Ricci; Niccolò Biondi; Federico Pernici; Alberto Del Bimbo

cs.CVOct 21, 2026

Hyperspherical Simplex Representations for Backward-Compatible Retrieval

Authors: Simone Ricci, Niccolò Biondi, Federico Pernici, Alberto Del Bimbo

Organizations: Media Integration and Communication Center (MICC) Dipartimento di Ingegneria dell’Informazione University of Florence · Multimedia & Human Understanding (MHUG) Department of Information Engineering and Computer Science University of Trento

Abstract

Training modern AI models has become increasingly expensive, and updating deployed models can alter their internal feature representations, affecting applications built upon them. In retrieval systems, such updates often require complete gallery re-indexing, a computationally expensive process at scale. Existing backward-compatibility methods enable direct comparison of representations from updated and earlier models, avoiding re-indexing. However, they typically require compatibility-specific training, learned mapping functions, or architectural modifications, and often depend explicitly on the previous model. We propose hyperspherical simplex representations derived from softmax outputs and logits for backward-compatible retrieval across independently trained models. They leverage the simplex structure induced by softmax cross-entropy training with one-hot labels as a common geometric reference across model updates. Since this reference depends on the number of classes, adding new ones expands the simplex, misaligning existing class prototypes with their earlier counterparts. We therefore derive a deterministic linear transformation to restore their alignment. Our theoretical analysis of the formal backward-compatibility definition identifies two geometric conditions under which its inequalities hold in expectation, assuming von Mises--Fisher class distributions with increasing concentration across updates. Aligned regular-simplex representations satisfy these conditions, providing a theoretical basis for the proposed representations under the stated assumptions. Without using additional compatibility-specific losses, learned mappings, or architectural modifications, the proposed representations achieve competitive results on standard compatibility benchmarks, including expanded class sets, more expressive architectures, and independently trained models from public repositories.

Figures & tables

Figure 1: Softmax outputs lie in the probability simplex $\Delta^{C^{k}-1}$ (colored area), centered at $\mathbf{o}^{k}$ , and are driven during training toward its vertices $\mathbf{e}_{c}$ , which correspond to the one-hot target labels. The projection $\mathbf{P}_{k,k}$ shifts the simplex center to the origin, mapping softmax outputs onto a regular simplex whose vertices define the class prototypes $\mathbf{v}^{k}_{c}$ (colored vectors). The features $\mathbf{h}^{k}$ of Eq. 4 are then obtained by $\ell_{2}$ -normalization onto the hypersphere, where each class is modeled by a vMF distribution with mean direction along the corresponding class prototype.

Figure 2: Softmax outputs and logits obtained from a ResNet-18 trained on three CIFAR-10 classes. While softmax outputs concentrate near the vertices of the probability simplex (colored area), logits exhibit a similar simplex geometry (gray area) centered at the origin with a broader within-class spread.

Figure 3: Plots of training loss and Neural Collapse (NC) hypotheses for softmax outputs (blue) and logits (orange) on CIFAR-100 with ResNet-18. The terminal phase of training is shown with a vertical red line.

Figure 4: Deterministic linear transformation across model updates of the class prototypes in Eq. 2 to which the hyperspherical simplex representations of Eq. 4 and Eq. 5 align. From left to right: Step 1, a 1-simplex with two classes (red and blue); Step 2, a 2-simplex with a new class (green); Step 3, a 3-simplex with a new class (yellow). Dashed arrows at steps 2 and 3 show how the class prototypes rotate to form the new simplex. The black dashed circle on the right shows the angular misalignment

\theta_{c}^{k\to t}

of the class-3 prototype between steps 2 and 3. Solid black arrows indicate the linear transformation

\mathbf{P}_{t,k}

, which aligns the updated prototypes of the previously existing classes with their counterparts at the earlier step.

Method	2 steps			5 steps			20 steps			50 steps
Method	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$
Baseline	0	29.63	0	0	13.60	0	0	04.22	0	0	02.29	0
BCT	1	40.64	35.51	0	31.03	0	0	22.66	0	0.01	15.78	0.84
RACT	1	44.19	45.09	0.60	30.43	16.53	0.05	21.61	2.42	0	16.71	0
AdvBCT	0	35.32	0	0.40	26.13	11.67	0.02	19.79	0.19	0	14.70	0.03
LCE	1	43.48	40.71	0.10	32.63	04.63	0	20.95	0.25	0	13.81	0.03

Table 1: Comparative analysis of compatibility performance under the “Extended Classes” benchmark on CIFAR-100 for multiple updates. Dark and light blue indicate respectively the highest and the second-highest value for each metric.

Method	ImageNet-1K						Google Landmark v2
	2 steps			5 steps			2 steps			5 steps
	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$
Baseline	0	37.52	0	0	16.18	0	0	07.66	0	0	07.87	0
BCT	1	48.81	45.86	0.20	34.96	5.28	1	09.37	08.22	0	08.46	0
RACT	1	55.17	51.16	0.10	40.94	6.20	0	11.01	0	0	09.99	0
AdvBCT	0	53.07	0	0	43.91	0	1	11.12	09.67	0	10.73	0

Table 2: Comparative analysis of compatibility performance under the “Extended Classes” benchmark on ImageNet-1K and Google Landmarks v2 for multiple updates. Dark and light blue indicate respectively the highest and the second-highest value for each metric.

Top- $k$	PSP			LSP
with $k=$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$
256	1	35.20	31.09	0.7	37.96	24.85
384	1	35.20	31.09	0.7	38.00	24.87
512	1	35.20	31.09	0.5	38.03	14.24
768	1	35.20	31.09	0.5	38.01	14.28
896	1	35.20	31.09	0.5	38.03	14.28

Table 3: Comparison of PSP and LSP across different top-

k

values in terms of

AC

AA

, and

ACA

, in the 5-step ImageNet-1K setting of the extended-classes benchmark.

Method	Ext. Backbone						Ext. Classes & Backbone
	CIFAR-100			ImageNet-1K			CIFAR-100			ImageNet-1K
	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$
Baseline	0	35.13	0	0	40.58	0	0	29.88	0	0	35.88	0
BCT	1	53.57	54.06	1	61.80	61.43	1	43.61	40.79	1	50.64	42.98
RACT	1	53.80	54.62	1	62.15	62.07	1	43.66	41.01	1	52.74	48.92
AdvBCT	0	49.56	0	1	59.22	57.73	1	39.06	35.05	1	50.37	45.42

Table 4: Comparative analysis of compatibility performance under the “Extended Backbone” and “Extended Classes and Backbone” benchmarks on CIFAR-100 and ImageNet-1K. Dark and light blue indicate respectively the highest and the second-highest value for each metric.

Method	AlexNet $\xrightarrow{}$ R-50 $\xrightarrow{}$ RegNet $\xrightarrow{}$ R-152 $\xrightarrow{}$ MaxViT						ViT-B-32 $\xrightarrow{}$ ViT-B-16 $\xrightarrow{}$ ViT-L-16
	ImageNet-1K			Places365			ImageNet-1K			Places365
	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$	$AC$	$AA$	$ACA$
Baseline	0	21.63	0	0	8.38	0	0	38.63	0	0	15.10	0
PSP	1	74.34	76.12	0.9	15.92	14.08	1	78.94	80.53	1	18.31	17.60
LSP	0.7	60.68	37.61	0.2	21.36	4.06	0.67	71.82	45.91	0.33	25.71	7.99

Table 5: Compatibility between pretrained models on ImageNet-1K and Places365: ( a ) AlexNet

\xrightarrow{}

ResNet-50 (R-50)

\xrightarrow{}

RegNetX_3.2GF (RegNet)

\xrightarrow{}

ResNet-152 (R-152)

\xrightarrow{}

MaxViT_T and ( b ) ViT-B-32

\xrightarrow{}

ViT-B-16

\xrightarrow{}

ViT-L-16. Models are pretrained on ImageNet-1K and downloaded from Marcel & Rodriguez (2010) . Dark blue numbers indicate the highest values, while light blue the second-highest values.

Panel of Figure 5 — Figure 5: Compatibility between independently trained models, all pretrained on ImageNet-1K. Compatibility matrices for five steps with increasingly expressive models (AlexNet, ResNet-50, RegNetX_3.2GF, ResNet-152, and MaxViT_T). The matrices display retrieval performance on the ImageNet-1K test set. Entries that do not satisfy the compatibility criterion in Eq. 8 are highlighted with a light-red background. Model representations from: (a) encoder output (Baseline); (b) softmax outputs (PSP); (c) logits (LSP).

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Plots of training loss and Neural Collapse hypotheses for feature representations derived from softmax outputs (blue) and logits (orange) on CIFAR-100 and Tiny-ImageNet-200 with ResNet-50 and DenseNet. The terminal phase of training is shown with a vertical red line.

Panel of Figure 7 — Figure 7: Expected cosine distances for aligned regular-simplex representations in the (a) same-class and (b) different-class cases, corresponding to Eq. 25a and Eq. 25b , respectively. The curves are obtained from the closed-form expected-distance expressions in Eq. 27a and Eq. 27b , under the assumptions of Theorem 3.6 , with the updated class distributions having higher concentration than the old ones. Across all considered feature-space dimensions $d$ , the expected distance between features from the updated and old models is lower than the corresponding old-model distance for same-class pairs and higher for different-class pairs, illustrating that the compatibility inequalities are satisfied in expectation.

Panel of Figure 8 — Figure 8: Per-class dispersion $\delta_{c}$ of the $\ell_{2}$ -normalized logits (a) and softmax outputs (b) on CIFAR-10 for the old model (solid bars) and the new model (hatched bars). In each subfigure, the left panel shows the extended-classes update from 5 to 10 classes, where the last 5 classes are unknown to the old model, and the right panel shows the update between independently trained models, with ResNet-18 as the old model and ResNet-50 as the new model, both pretrained on ImageNet-1K. Lower values of $\delta_{c}$ indicate a tighter concentration around the class mean direction.

Method	$\mathbf{P}_{k,k}$	$\mathbf{P}_{t,k}$	$\gamma_{\mathrm{same}}$	$\gamma_{\mathrm{diff}}$	Compatible pairs
$\ell_{2}$ -Softmax	–	–	$-0.0111\pm 0.0030$	$-0.0055\pm 0.0009$	0/9
PSP w/o $\mathbf{P}_{t,k}$	✓	–	$-0.0568\pm 0.0033$	$-0.0142\pm 0.0008$	0/9
PSP w/o $\mathbf{P}_{k,k}$	–	✓ ∗	$+0.0120\pm 0.0019$	$-0.0133\pm 0.0010$	0/9
PSP	✓	✓	$+0.0155\pm 0.0025$	$+0.0039\pm 0.0006$	9/9

Appendix

Table 6: Analysis of the centering projection

\mathbf{P}_{k,k}

and the deterministic transformation

\mathbf{P}_{t,k}

on CIFAR-10 with ResNet-18 under class expansion (

5\to 10

classes). Average margins of Eq. 28 (mean

\pm

std over 9 model pairs; positive values indicate that the inequality is satisfied on average) and number of compatible pairs, i.e., model pairs for which both margins are positive. ∗ Without the centering projection, i.e.,

[\mathbf{I}_{C^{k}}\,|\,\mathbf{0}]

in place of

\mathbf{P}_{t,k}

. Dark and light blue indicate respectively the highest and the second-highest value for each column.

Panel of Figure 9 — Figure 9: Compatibility matrices showing CMC@ $1$ retrieval metric values at each model update step for the 5-step scenario on the CIFAR-100 dataset. Each matrix entry failing to satisfy the compatibility criterion defined in Eq. 8 is highlighted with a light-red background.

Panel of Figure 10 — Figure 10: Open-set scenario. Compatibility matrices for five steps with increasingly expressive models (AlexNet, ResNet-50, RegNetX_3.2GF, ResNet-152, and MaxViT_T), all pretrained on ImageNet-1K. The matrices display retrieval performance on the Places365 dataset. Entries that do not satisfy the compatibility criterion in Eq. 8 are highlighted with a light-red background. Model representations obtained from: (a) encoder output (Baseline); (b) softmax outputs (PSP); (c) logits (LSP).

Panel of Figure 11 — Figure 11: Closed-set scenario. Compatibility matrices for three steps with advanced Vision Transformer (ViT) model architectures (ViT-B/32, ViT-B/16, and ViT-L/16), all pretrained on ImageNet-1K. The matrices show closed-set retrieval performance on the ImageNet-1K test set. Entries that do not satisfy the compatibility criterion in Eq. 8 are highlighted with a light-red background. Model representations obtained from: (a) encoder output (Baseline); (b) softmax outputs (PSP); (c) logits (LSP).

Panel of Figure 12 — Figure 12: Open-set scenario. Compatibility matrices for three steps with advanced Vision Transformer (ViT) model architectures (ViT-B/32, ViT-B/16, and ViT-L/16), all pretrained on ImageNet-1K. The matrices show open-set retrieval performance on the Places365 test set. Entries that do not satisfy the compatibility criterion in Eq. 8 are highlighted with a light-red background. Model representations obtained from: (a) encoder output (Baseline); (b) softmax outputs (PSP); (c) logits (LSP).

Table 7: Compatibility results for self-supervised and vision-language pretrained encoders in a linear-probe setting on ImageNet-1K. For each checkpoint, a linear probe is trained on top of frozen backbone features. Dark blue numbers indicate the highest values, while light blue numbers indicate the second-highest values.

Panel of Figure 13 — Figure 13: Compatibility matrices for three steps with self-supervised Vision Transformer checkpoints from the DINO family (DINO-ViT-S, DINOv2-ViT-S/14, and DINOv3-ViT-L/16) obtained from Hugging Face Hub ( Wolf et al., 2019 ) . For each checkpoint, we train a linear probe on ImageNet-1K with the backbone frozen, and construct PSP/LSP features from the resulting classifier outputs. The matrices show closed-set retrieval performance on the ImageNet-1K test set. Entries that do not satisfy the compatibility criterion in Eq. 8 are highlighted with a light-red background. Model representations obtained from: (a) encoder output (Baseline); (b) softmax outputs (PSP); (c) logits (LSP).

Panel of Figure 14 — Figure 14: Compatibility matrices for three steps with vision-language CLIP family (ViT-B/32, ViT-B/16, and ViT-L/14) pretrained by OpenAI. For each checkpoint, we train a linear probe on ImageNet-1K with the backbone frozen, and construct PSP/LSP features from the resulting classifier outputs. The matrices show closed-set retrieval performance on the ImageNet-1K test set. Entries that do not satisfy the compatibility criterion in Eq. 8 are highlighted with a light-red background. Model representations obtained from: (a) encoder output (Baseline); (b) softmax outputs (PSP); (c) logits (LSP).

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