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Intrinsic Dimension Correlation: uncovering nonlinear connections in multimodal representations
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Proposes Intrinsic Dimension Correlation (I_dCor), a normalized-mutual-information-style correlation metric that replaces entropy with TwoNN intrinsic dimension estimates, and uses it to reveal nonlinear correlations between paired image and text encoder representations where CKA, SVCCA and distance correlation fail.
Problem
Standard representation similarity metrics (CKA, SVCCA, distance correlation) fail to detect strongly nonlinear, high-dimensional correlations, e.g., between paired visual and textual embeddings from different encoders.
Approach
Define a correlation coefficient as normalized mutual information in which entropy is replaced by intrinsic dimension: (I_d(X)+I_d(Y)-I_d(X⊕Y))/max(I_d(X),I_d(Y)), with a permutation-test p-value on the joint intrinsic dimension.
Key results
- On the spiral toy dataset, I_dCor reaches 0.98 correlation with p=0.01 while R^2 and distance correlation are near 0.
- On N24News, I_dCor yields mean off-diagonal correlation 0.66 versus 0.25-0.29 for baselines, detecting cross-modal correlations that baselines miss.
- On ImageNet coarse alignment, I_dCor averages 0.62 vs. 0.31-0.47 for baselines, with p-values of 0.01 for all models.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use I_dCor to measure the spiral variables' dependence | two-variable synthetic spiral data · Spiral | correlation coefficient: 0.01±0.00 (R²) / 0.02±0.00 (dCor) -> 0.98±0.02 | Helped |
| use I_dCor rather than CCA or dCor | four-variable synthetic data · Partial correlation | correlation coefficient: 0.02±0.01 (CCA), 0.00±0.00 (dCor) -> 0.35±0.03 | Helped |
| use I_dCor rather than CCA or dCor | four-variable synthetic data · Complete nonlinear correlation | correlation coefficient: 0.02±0.00 (CCA), 0.01±0.00 (dCor) -> 0.67±0.02 | Helped |
| increase MLP activation nonlinearity by lowering the LeakyReLU slope | I_dCor on MNIST input and final-layer representations · MNIST | correlation coefficient | No clear effect |
| increase MLP activation nonlinearity by lowering the LeakyReLU slope | SVCCA, dCor, linear CKA and RBF CKA · MNIST | correlation coefficient | Hurt |
| use I_dCor instead of dCor on aligned representations | ImageNet encoders · ImageNet representations | mean off-diagonal correlation: 0.50 -> 0.87 | Helped |
| use I_dCor instead of baseline similarity metrics | ImageNet representations under coarse class-preserving alignment · ImageNet coarse alignment | average correlation: 0.31 (SVCCA), 0.42 (CKA), 0.47 (dCor) -> 0.62 | Helped |
| freely shuffle one representation without preserving class labels | ImageNet representations measured with I_dCor · ImageNet random shuffling | correlation significance: p-value [0.08, 0.99] | No clear effect |
| use I_dCor instead of dCor for cross-modal representations | N24News image and text encoders · N24News | mean off-diagonal correlation: 0.25-0.29 -> 0.66 | Helped |
| replace TwoNN with MLE for intrinsic-dimension estimation (ablation) | I_dCor on four-variable synthetic data · Complete nonlinear correlation | I_dCor: 0.67±0.02 -> 0.38±0.01 | Hurt |
| replace TwoNN with MLE for intrinsic-dimension estimation (ablation) | I_dCor on two-variable synthetic data · Spiral | I_dCor: 0.98±0.02 -> 1.06±0.00 | Mixed |
| sample the original four-variable data uniformly in [-π, π] instead of from a Gaussian | I_dCor · Four-variable synthetic data: complete nonlinear correlation | I_dCor: 0.67±0.02 -> 0.92±0.02 | Helped |
Surprising findings
- Contrastive vision-language encoders (CLIP, BLIP, SigLIP) showed lower I_dCor under coarse alignment, sometimes below distance correlation and CKA.
- The metric is largely insensitive to the nonlinearity of the transformation, so it is a less informative geometric similarity index than CKA or dCor.
Limitations
- Depends entirely on the accuracy of the intrinsic dimension estimator, which fails at large intrinsic dimensions (curse of dimensionality).
- Only applicable to datasets with uniform intrinsic dimension in its present form.
- Insensitive to nonlinear transformations, so it does not describe geometric similarity well.
Experimental setup
Datasets: MNIST ImageNet N24News MS-COCO 2014 Flickr30k
Benchmarks: MNIST ImageNet N24News MS-COCO 2014 Flickr30k
Metrics: I_dCor correlation coefficient p-value mean off-diagonal correlation
Baselines: SVCCA PWCCA linear CKA RBF CKA Distance Correlation CCA RTD
Builds on: normalized mutual information TwoNN intrinsic dimension estimator CKA distance correlation
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