Mutual Information (MI) is a fundamental measure of statistical dependence widely used in representation learning. While direct optimization of MI via its definition as a Kullback-Leibler divergence (KLD) is often intractable, many recent methods have instead maximized alternative dependence measures, most notably, the JensenShannon divergence (JSD) between joint and product of marginal distributions via discriminative losses. However, the connection between these surrogate objectives and MI remains poorly understood.

Generative neural networks learn how to produce highly realistic images from a large, but finite number of examples - or do they simply memorise their training set? To settle this question, Kadkhodaie, Guth, Simoncelli and Mallat (ICLR '24) trained diffusion models independently on disjoint subsets of a dataset and showed that they converge to nearly the same density when the number of training images is large enough. This result raises two basic questions: how much data do you need for convergence, and what does convergence capture about learning the data distribution? Here, we address these questions by providing an exact analytical characterisation of the transition from memorisation to generalisation in linear generative models. We find that these models memorise at small load, while convergence emerges continuously when the number of samples is linear in the input dimension. Strikingly, we find that convergence is insensitive to recovery of the principal latent factors of the data, which are recovered in a sharp transition. After extending our approach to data with power-law spectra, we find the same distinction between convergence and latent recovery in our experiments with convolutional denoisers and in the data of Kadkhodaie et al. We thus show that generalisation in generative models decomposes into at least two distinct objectives: matching the bulk of the data distribution and recovering the principal latent factors. These objectives correspond to two different distances between true and learnt data distribution, and only the first one is captured by convergence.

Since pioneering work of Hinton et al., knowledge distillation based on Kullback-Leibler Divergence (KL-Div) has been predominant, and recently its variants have achieved compelling performance. However, KL-Div only compares probabilities of the corresponding category between the teacher and student while lacking a mechanism for cross-category comparison. Besides, KL-Div is problematic when applied to intermediate layers, as it cannot handle non-overlapping distributions and is unaware of geometry of the underlying manifold. To address these downsides, we propose a methodology of Wasserstein Distance (WD) based knowledge distillation. Specifically, we propose a logit distillation method called WKD-L based on discrete WD, which performs cross-category comparison of probabilities and thus can explicitly leverage rich interrelations among categories. Moreover, we introduce a feature distillation method called WKD-F, which uses a parametric method for modeling feature distributions and adopts continuous WD for transferring knowledge from intermediate layers. Comprehensive evaluations on image classification and object detection have shown (1) for logit distillation WKD-L outperforms very strong KL-Div variants; (2) for feature distillation WKD-F is superior to the KL-Div counterparts and state-of-the-art competitors.

From the three-points identity of the Bregman divergence (Lemma 3.1 of [9]), KL (x y) KL ( x y) = KL (x x) + ln x ln y,x x (12) The first term in (12) can be bounded by KL (x x) = By the Hölder's inequality, the second term in (12) is bounded as ln x ln y,x x ln x ln y Lemma 5. Consider a block diagonal matrix We prove the lemma via induction on N . This completes the induction proof.Lemma 6. We introduce one more notation before presenting the proof. This leads us to the initialization-dependent convergence rate of Algorithm 1, which we re-state and prove as follows. In addition, if we initialize the players' policies to be uniform policies, i.e., The rest of the proof follows by putting all the aforementioned results together.

Our theoretical analysis reveals that in this setting, in-context learning is more about identifying the task than about learning it, a result which is in line with a series of recent empirical findings.

Discovering clusters in bipartite graphs has been researched in many different settings. However, most of these algorithms were heuristics and do not provide theoretical guarantees for the quality oftheir results.

Kullback-Leibler (KL) divergence is one of the most important measures to calculate the difference between probability distributions. In this paper, we theoretically study several properties of KL divergence between multivariate Gaussian distributions.