We start by providing motivation for the unconstrained Jacobians problem introduced in the main text. We will continue our proof using contradiction. Figure 1: Fully-connected ResNet34 (Type 1 model) trained on MNIST.Figure 2: Fully-connected ResNet34 (Type 1 model) trained on FashionMNIST. Figure 10: Fully-connected ResNet34 (Type 1 model) trained on MNIST. Figure 24: Fully-connected ResNet34 (Type 1 model) trained on MNIST.

The ResNet architecture has been widely adopted in deep learning due to its significant boost to performance through the use of simple skip connections, yet the underlying mechanisms leading to its success remain largely unknown. In this paper, we conduct a thorough empirical study of the ResNet architecture in classification tasks by linearizing its constituent residual blocks using Residual Jacobians and measuring their singular value decompositions.

Starting from flow- and diffusion-based transformers, Multi-modal Diffusion Transformers (MM-DiTs) have reshaped text-to-vision generation, gaining acclaim for exceptional visual fidelity. As these models advance, users continually push the boundary with imaginative or rare prompts, which advanced models still falter in generating, since their concepts are often too scarce to leave a strong imprint during pre-training. In this paper, we propose a simple yet effective intervention that surfaces rare semantics inside MM-DiTs without additional training steps, data, denoising-time optimization, or reliance on external modules (e.g., large language models). In particular, the joint-attention mechanism intrinsic to MM-DiT sequentially updates text embeddings alongside image embeddings throughout transformer blocks. We find that by mathematically expanding representational basins around text token embeddings via variance scale-up before the joint-attention blocks, rare semantics clearly emerge in MM-DiT's outputs. Furthermore, our results generalize effectively across text-to-vision tasks, including text-to-image, text-to-video, and text-driven image editing. Our work invites generative models to reveal the semantics that users intend, once hidden yet ready to surface.

Whitening is a classical technique in unsupervised learning that can facilitate estimation tasks by standardizing data. An important application is the estimation of latent variable models via the decomposition of tensors built from high-order moments. In particular, whitening orthogonalizes the means of a spherical Gaussian mixture model (GMM), thereby making the corresponding moment tensor orthogonally decomposable, hence easier to decompose. However, in the large-dimensional regime (LDR) where data are high-dimensional and scarce, the standard whitening matrix built from the sample covariance becomes ineffective because the latter is spectrally distorted. Consequently, whitened means of a spherical GMM are no longer orthogonal. Using random matrix theory, we derive exact limits for their dot products, which are generally nonzero in the LDR. As our main contribution, we then construct a corrected whitening matrix that restores asymptotic orthogonality, allowing for performance gains in spherical GMM estimation.

The ResNet architecture has been widely adopted in deep learning due to its significant boost to performance through the use of simple skip connections, yet the underlying mechanisms leading to its success remain largely unknown. In this paper, we conduct a thorough empirical study of the ResNet architecture in classification tasks by linearizing its constituent residual blocks using Residual Jacobians and measuring their singular value decompositions. It also provably occurs in a novel mathematical model we propose. This phenomenon reveals a strong alignment between residual branches of a ResNet (RA2 4), imparting a highly rigid geometric structure to the intermediate representations as they progress *linearly* through the network (RA1) up to the final layer, where they undergo Neural Collapse.