PHI-S: Distribution Balancing for Label-Free Multi-Teacher Distillation

TL;DR

This work addresses unlabeled multi-teacher distillation by examining how activation statistics and loss functions shape the student. It introduces PHI-S, an isotropic standardization based on PCA-Hadamard rotation, and systematically compares standardization, whitening, and Hadamard-based methods across diverse teacher distributions. PHI-S consistently yields superior or competitive performance on a range of tasks, including segmentation and multimodal probes, and offers insights into error balance and rank stability. The findings suggest invertible, isotropic normalization as a practical strategy for robust, label-free multi-teacher distillation at scale.

Abstract

Various visual foundation models have distinct strengths and weaknesses, both of which can be improved through heterogeneous multi-teacher knowledge distillation without labels, termed "agglomerative models." We build upon this body of work by studying the effect of the teachers' activation statistics, particularly the impact of the loss function on the resulting student model quality. We explore a standard toolkit of statistical normalization techniques to better align the different distributions and assess their effects. Further, we examine the impact on downstream teacher-matching metrics, which motivates the use of Hadamard matrices. With these matrices, we demonstrate useful properties, showing how they can be used for isotropic standardization, where each dimension of a multivariate distribution is standardized using the same scale. We call this technique "PHI Standardization" (PHI-S) and empirically demonstrate that it produces the best student model across the suite of methods studied.

Figures (9)

• Figure 1: Illustration of the modified agglomerative model training procedure. Instead of the student model learning to match the original teacher distributions, it learns to match the normalized distributions (our proposed PHI-S is shown). We show the real distributions for DFN CLIP, DINOv2, SigLIP, and SAM by projecting them down to 2D using PCA. In the original space, the variance of DFN CLIP is so small that it appears as a single point. During inference, we can estimate the original teacher distributions using the inverse normalization process on the student predictions.
• Figure 2: Teacher activation histograms. We show the global histogram, as well as the histograms for the channels associated with the minimum mean, maximum mean, minimum variance, and maximum variance. While all being roughly normal, they have very different centers and scales. We provide specific values in table \ref{['tab:teacher_activation_stats']} in the appendix.
• Figure 3: The loss curves for each of the four teachers that the ViT-B/16 student is learning to match simultaneously in original teacher space (e.g. denormalized). We emphasize "Baseline - MSE" (Blue) and "PHI Standardize" (PHI-S, Red) as they generally set the upper and lower bounds.
• Figure 4: Visualization of how standardization affects the resulting data distribution. We start with the same distribution, and rotate the data by some angle. Regular standardization's effect is directly tied to the distribution orientation. Conversely, PHI-S is invariant to any data rotation, and will produce an identical transform up to sign along each dimension. We can make the sign consistent by negating the rows of $\mathbf{H}$ and $\mathbf{U}$ which have a negative value in the diagonal position. Similarly, regular standardization will distort each dimension (shown with red/blue lines), which will have the effect of reducing the importance of high variance axis-aligned dimensions, and increasing the importance of low-variance dimensions. PHI-S is isotropic, so the change in scale is uniform.
• Figure 5: Visualization of normalization procedures. We display two axis lines in red and blue. In the original space, they're both 2 units long, and aligned with the plot coordinate system. We also display an "error circle" which is a unit circle in the normalized coordinate system. For the three whitening transforms you can see how they only differ by rotation. We also specifically draw colored dots on the error circle corresponding to the extremal points of the error circle when denormalized into an ellipse. PCA-W places the largest error magnitude on the x-axis, given that it's the dimension with largest eigenvalue thus estimation errors along the x dimension will have a much larger impact in the denormalized space. As we show in equation \ref{['eq:error_zca']}, the error for ZCA will be proportional to the original distribution's covariance matrix, and thus, the extremal points are along the eigenvectors of the covariance matrix. Hadamard whitening has the extremal points at $\left|x_1\right| = \left|x_2\right| = ... = \left|x_C\right|$. Global Standardization and PHI-S are both isotropic, which means that there's an infinite number of extremal points, so we instead show the points as they relate to the distribution itself. Similar to ZCA, for Global Standardization these points are along the principal axes. And similar to HCA, the aligned points for PHI-S are when $\left|x_1\right| = \left|x_2\right| = ... = \left|x_C\right|$.

Theorems & Definitions (2)

• Theorem 2.1
• proof