Spherical Interpolation for Backward-Compatible Multimodal Representations

Simone Ricci; Niccolò Biondi; Federico Pernici

cs.CVSep 30, 2026

Spherical Interpolation for Backward-Compatible Multimodal Representations

Authors: Simone Ricci, Niccolò Biondi, Federico Pernici

Organizations: DINFO (Department of Information Engineering), University of Florence, Italy · MICC (Media Integration and Communication Center) · University of Trento, Italy

Abstract

Contrastive vision-language models map visual and textual representations into a shared normalized embedding space, making cosine similarity the natural metric for cross-modal retrieval. A practical challenge arises during model upgrades: independently trained models generally produce incompatible representation spaces, so replacing a deployed model typically requires recomputing embeddings for the entire gallery, which is prohibitively expensive at scale. Orthogonal post-hoc alignment can partially mitigate this problem by mapping new-model queries into the old-model gallery space. However, because independently trained models can differ in fine-grained representation structure, the orthogonal alignment remains approximate, leaving a residual angular discrepancy between the old-model query and the aligned new-model query. We study whether interpolation along the spherical geodesic between these two normalized query representations can improve retrieval without re-indexing the gallery. We characterize when this path contains an interior query direction closer to an idealized retrieval-optimal direction than either endpoint, and connect this characterization to Recall@ $K$ through a local margin-based certification result. Experiments across multiple benchmarks and model families show that post-alignment spherical interpolation improves over orthogonal alignment alone, recovering backward-compatibility in most evaluated settings. Consistent with our geometric characterization, per-query oracle analysis shows that retrieval-favorable interior points occur frequently in practice. Code is available at https://github.com/miccunifi/SLERP_backward_compatibility .

Figures & tables

Panel of Figure 1 — Figure 1 : Residual discrepancy after orthogonal alignment from SigLIP2 to CLIP ViT-B/32. We show two complementary views: the distribution of residual angular mismatch between paired embeddings from the two models, and a 2D t-SNE projection of the corresponding feature spaces. The remaining angular mismatch after alignment suggests that the orthogonal map leaves a residual cross-model gap, motivating a post-alignment interpolation strategy.

Panel of Figure 2 — Figure 2 : Geometry of SLERP in the interpolation plane. The old query $u$ and aligned new query $v$ define a minor geodesic arc. Let $p$ be the projection of the retrieval-optimal direction $q^{\ast}$ onto this plane, and let $\rho=\|p\|$ . The effect of interpolation is determined by the normalized in-plane projection $p/\rho$ when $\rho>0$ . A strict interior improvement occurs exactly when $p/\rho$ lies in the relative interior of the arc; when $\rho=0$ , the angular gap is constant along the arc.

Panel of Figure 3 — Figure 3 : Text-to-image Recall@1 along the SLERP interpolation path for Flickr30k (left) and COCO (right). The image gallery is encoded by the old model, while text queries interpolate between the old-model query ( $\alpha=0$ ) and the SVD-aligned new-model query ( $\alpha=1$ ). Markers indicate the CC3M-selected $\hat{\alpha}$ and the dataset-specific oracle $\alpha^{\star}$ ; dashed lines report the old-model, new-model, and SVD performance.

Panel of Figure 4 — Figure 4 : Flip-rate trade-off for T2I Recall@1. Bars show positive and negative flip rates relative to the old-to-old evaluation. The blue curve reports $\Delta\mathrm{R@1}$ . Here $\alpha=0$ corresponds to old-model queries and $\alpha=1$ to SVD-aligned new-model queries. Red and blue markers denote the dataset-specific oracle $\alpha^{\star}$ and the CC3M-selected weight $\hat{\alpha}$ , respectively.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5 : Empirical comparison of SLERP and NLERP for text-only image-to-text retrieval after SVD alignment. The columns correspond to the two source-target model pairs: CLIP ViT-L/14 $\rightarrow$ CLIP ViT-B/32 on the left and SigLIP2 $\rightarrow$ CLIP ViT-B/32 on the right. The rows correspond to the two retrieval datasets: Flickr30k on the top row and COCO on the bottom row. Across Flickr30k and COCO, and for both source-model pairs, the two interpolation schemes trace nearly identical retrieval curves and attain their best performance in the same interior region of the interpolation path. The small discrepancies arise because a uniform grid in the NLERP weight does not correspond to a uniform grid in angular position along the shared geodesic arc. This supports using SLERP as a geometrically canonical parameterization rather than as a different family of query representations.

Panel of Figure 6 — Figure 6 : Image-to-text Recall@1 along the SLERP path for CLIP ViT-H/14 $\rightarrow$ CLIP ViT-B/32. The text gallery is encoded by the old model, while image queries interpolate between the old-model query and the SVD-aligned new-model query. The case $\alpha=0$ corresponds to old-model queries, while $\alpha=1$ corresponds to SVD-aligned new-model queries. Markers show the CC3M-selected $\hat{\alpha}$ and the dataset-specific test oracle $\alpha^{\star}$ ; dashed lines show old-model, SVD, and new-model reference performance.

Panel of Figure 7 — Figure 7 : Image-to-text Recall@1 along the SLERP path for SigLIP1 $\rightarrow$ CLIP ViT-H/14. The text gallery is encoded by the old model, while image queries interpolate between the old-model query and the SVD-aligned new-model query. The case $\alpha=0$ corresponds to old-model queries, while $\alpha=1$ corresponds to SVD-aligned new-model queries. Markers show the CC3M-selected $\hat{\alpha}$ and the dataset-specific test oracle $\alpha^{\star}$ ; dashed lines show old-model, SVD, and new-model reference performance.

Figure 8 : Recall@1 along the SLERP path under different target-set fractions. Solid curves report the average over three random seeds; shaded regions denote $\pm 1\sigma$ across seeds.

Panel of Figure 9 — Figure 9 : I2T Recall@1 flip-rate trade-off on Flickr30k and COCO. Bars show positive and negative flip rates relative to the old-to-old evaluation. The blue curve reports $\Delta\mathrm{R@1}=\mathrm{PFR}-\mathrm{NFR}$ . Here $\alpha=0$ corresponds to old-model queries and $\alpha=1$ to SVD-aligned new-model queries. Red and blue markers denote the dataset-specific oracle $\alpha^{\star}$ and the CC3M-selected weight $\hat{\alpha}$ , respectively.

Panel of Figure 10 — Figure 10 : NoCaps Recall@1 flip-rate trade-off. We report both T2I and I2T flip rates for the same old–new pairs used in the main paper. Bars show positive and negative flip rates relative to the old-to-old evaluation, and the blue curve reports $\Delta\mathrm{R@1}=\mathrm{PFR}-\mathrm{NFR}$ . Here $\alpha=0$ corresponds to old-model queries and $\alpha=1$ to SVD-aligned new-model queries. Red and blue markers denote the dataset-specific oracle $\alpha^{\star}$ and the CC3M-selected weight $\hat{\alpha}$ , respectively.

Panel of Figure 11 — Figure 11 : Per-query oracle analysis on Flickr30k image-to-text retrieval for CLIP ViT-L/14 $\rightarrow$ CLIP ViT-B/32 with text-only alignment. (a) Distribution of per-query oracle interpolation weights, where each query is assigned the weight attaining the best rank of its highest-ranked relevant item, with ties broken by the highest relevant-item similarity and then by proximity to $\alpha=0.5$ . Queries are divided into those for which Recall@1 is achievable at some interpolation weight (blue, $n=18{,}344$ ) and those for which it is not (gray, $n=12{,}670$ ); shaded columns mark the two endpoints of the interpolation path. (b) Recall@1 obtained with a single fixed interpolation weight applied to all queries, maximized at the dataset-level optimum $\alpha^{\star}=0.5$ . (c) Cosine similarity between the interpolated query and its highest-scoring relevant item at each weight, averaged over Recall@1-achievable queries. Red dashed and cyan dotted lines mark the dataset oracle $\alpha^{\star}=0.5$ and the CC3M-selected weight $\hat{\alpha}=0.7$ , respectively. Because most successful queries attain rank one over many weights, the oracle distribution follows the relevant-item similarity and peaks at $\alpha=0.3$ , rather than at the Recall@1-optimal fixed weight $\alpha^{\star}$ .

Method	R@1
$\blacktriangleright$ Old model	40.62
SVD	42.89
SLERP ( $\hat{\alpha}=0.7$ )	48.00
SLERP ( $\alpha^{\star}=0.5$ )	48.61
$\blacktriangleright$ New model	48.72
Endpoints only (per-query oracle)	54.61

Appendix

Table 9 : Flickr30k I2T Recall@1 (%) for CLIP ViT-L/14

\rightarrow

ViT-B/32 with text-only alignment.

Method	I2T	T2I	$\Delta_{\mathrm{SVD}}$	$\Delta_{\mathrm{old}}$	Comp.
$\blacktriangleright$ Old-model query	53.36	34.26	+1.06	0.00	–
Alignment only
SVD (Procrustes)	51.72	33.77	0.00	$-1.06$	45/90
Affine	31.69	23.10	$-15.35$	$-16.42$	11/90
Ridge	33.96	28.89	$-11.32$	$-12.38$	14/90
One-sided CCA	33.96	28.89	$-11.32$	$-12.38$	14/90

Appendix

Table 10 : Aggregate Recall@1 ablation over

45

configurations (

90

compatibility evaluations) reported in Tabs. 4 , 5 , and 6 .

\Delta_{\mathrm{SVD}}

and

\Delta_{\mathrm{old}}

are averaged over both retrieval directions; Comp. counts evaluations satisfying Eq. 15 .

			Encoder compute (GFLOPs)		Latency (ms)
Serving configuration	Gallery	Query encoders	I2T	T2I	I2T	T2I
Old model	$\phi_{\mathrm{old}}$	$\phi_{\mathrm{old}}$	8.82	5.96	19.93	19.95
Full re-indexing	$\phi_{\mathrm{new}}$	$\phi_{\mathrm{new}}$	162.03	13.30	25.59	20.40
SLERP, one GPU	$\phi_{\mathrm{old}}$	$\phi_{\mathrm{old}}+\phi_{\mathrm{new}}$	170.85	19.26	31.04	25.76
SLERP, two GPUs	$\phi_{\mathrm{old}}$	$\phi_{\mathrm{old}}+\phi_{\mathrm{new}}$	170.85	19.26	25.87	20.70
SLERP, cached old query	$\phi_{\mathrm{old}}$	$\phi_{\mathrm{new}}$	162.03	13.30	25.87	20.68

Appendix

Table 11 : Per-query serving cost for CLIP ViT-L/14

\rightarrow

CLIP ViT-B/32 with a gallery of

10^{7}

items on one NVIDIA A100. Encoder compute is taken from the OpenCLIP profiles and excludes the Procrustes projection (

\approx 0.79

MFLOPs) and the search. Latency is the sum of independently measured median (p50) encoder and IVF-PQ search components; in the two-GPU setting, the old and new encoders are executed concurrently on separate devices.

Figure 12 : Image-to-text Recall@1 during partial backfilling of the text gallery on Flickr30k, with text-only Procrustes support. The partial-backfill baseline issues SVD-aligned new-model queries against a mixed gallery in which an increasing fraction of old-model caption embeddings is replaced by new-model embeddings. At

0\%

, the baseline coincides with the SVD query on the fixed old-model gallery (square), and at

100\%

with full re-indexing (star). SVD+SLERP uses the CC3M-selected

\hat{\alpha}

and leaves the gallery unchanged. Shaded regions mark backfilling fractions at which the baseline falls below the old-model system and thus violates Eq. 15 .

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