data compression teams

Abstract: Motivated by the notion of features from machine learning, we study the complexity of representing data that allows for evaluation with a given family of functions. Concretely, we consider a deterministic encoder that observes $X\sim D$ and produces a finite encoding before a task $f\in\mathcal{F}\subseteq\{0,1\}^{\mathcal{X}}$ is specified. For each task, a decoder must predict $f(X)$ with error at most $\varepsilon$ under $D$. We construct an encoder that outputs a representation with $O(d^{*}\log(1/\varepsilon))$ bits, where $d^{*}$ is the dual VC dimension of $\mathcal{F}$. Our encoder greedily selects $O((d^{*}/\varepsilon)\log(1/\varepsilon))$ functions from the class and uses their joint label pattern to produce the encoding. The key idea in our upper bound is a novel potential function argument, where we compare the entropy gained from each greedy selection with the number of patterns allowed by the dual VC dimension. We present lower bounds showing that the representation length achieved by our encoder is worst-case optimal for every $d^* \ge 1$ and $\varepsilon < 1/4$ up to a universal constant factor, and instance-wise optimal for every family $\mathcal{F}$ up to an $O(\log(1/\varepsilon))$ factor. We give two ways to learn the representation from unlabeled samples, with different guarantees on the representation size and sample complexity. Experimentally, we verify the greedy construction's accuracy-size tradeoff and explore its connection to neural sparse autoencoders.

Abstract: Expand-and-sparsify ($\texttt{EaS}$) representations lift an input to a much higher dimension through a linear map and keep only the $k$ largest coordinates. Existing theory builds that map at random and needs the expanded dimension to grow exponentially in the input dimension, which puts the construction out of practical reach. We ask what is gained by learning the map by gradient descent instead, and how that gain depends on how wide the expansion is. We sweep $73$ OpenML classification tasks and $48$ regression tasks under a protocol that holds the datasets, expanded dimensions, sparsity levels and splits fixed across every method, and we summarise each comparison by a median over datasets and a Wilcoxon signed rank test. We report four findings. First, learning the expansion helps most when the expansion is narrowest, and its advantage over a random map decays to nothing by the widest expansions we run, so a learned $\texttt{EaS}$ layer can be made small. Second, with the map learned, a top-$k$ representation that keeps at most a tenth of its coordinates is indistinguishable from a dense $\texttt{ReLU}$ layer of the same expanded dimension at every expansion factor we test, while GELU and SiLU are not. Third, when the representation is forced to be binary the sparse code is the better of the two, and the reason is that binarization costs it nothing while it costs the dense layer accuracy. Fourth, how sparse the representation can be is governed by how wide it is: the sparsity level that is free rises as the expansion grows, so there is no fixed limit on how sparse it can be. The regression tasks reproduce the first finding and qualify the second and third.

Abstract: Discrete diffusion models have emerged as a leading alternative to autoregressive models for generating text and other categorical data, yet their theoretical foundations remain less developed than those of their continuous counterparts. For continuous diffusion models, a prominent line of theory suggests that these models can adapt to low-dimensional structure in the data, thereby improving statistical efficiency and generalization. In this work, we investigate the discrete analogue of this phenomenon by studying the statistical rates of discrete diffusion models under sparsity assumptions. By analyzing a nonparametric estimator derived from a regularized score-entropy objective, we establish minimax-optimal rates in the sparse regime. The proposed regularization can be viewed as an adaptation of label smoothing techniques introduced for classification, and does not require the knowledge of the sparsity parameter. This shows that discrete diffusion models can adapt intrinsically to sparsity.

Abstract: Madiman and Tetali established strong and weak fractional inequalities that subsume classical entropy subset inequalities such as Han's inequality and Shearer's lemma. In this work, we derive a decomposition of the gap in the strong fractional inequality and use it to establish the necessary and sufficient conditions for equality in the strong form of Han's inequality. As a consequence, we characterize the equality conditions for a strong Loomis-Whitnet-type projection inequality of Balister and Bollobas (2012). Finally, we apply the weak fractional inequality to an extremal graph theory problem, recovering and extending results of Sason (2022) and Boucheron et al. (2013), whose approaches rely on Han's inequality.

Abstract: We study the communication cost of exact sampling under Campbell's exponential coding criterion. Given a target distribution $P$ and a proposal distribution $Q$, the sender and receiver share i.i.d. $Q$-distributed samples, and the sender communicates an index so that the selected sample has distribution exactly $P$. We introduce a new information measure Rényi channel simulation divergence $R_α(P\Vert Q)$, a Rényi generalization of the channel simulation divergence. We then show that it lower bounds the Campbell cost of every exact sampler and is achievable, via a new layered sampling scheme, within an additive constant depending only on the Campbell parameter. This yields a tighter one-shot characterization than previous bounds based on Rényi divergences of different orders. For i.i.d. product distributions, under suitable non-degeneracy and moment conditions, we further show that the gap beyond the leading Rényi-divergence term is $\frac{1}{2}\log n+O(1)$.

Abstract: This work studies an entropy-based estimator of Kullback-Leibler (KL) divergence constructed from normalized importance weights. We prove a universal finite-sample bound showing that its expectation can exceed the target-proposal KL divergence by at most $\log 2$ for any sample size. We then characterize the KL divergence between the distribution induced by importance-weighted resampling and the proposal, showing that it is upper bounded by both the expectation of the entropy-based estimator and the target-proposal KL divergence. Importantly, these bounds require neither bounded importance weights nor finite higher moments. We apply these results to test-time alignment of generative models via Soft Best-of-$N$ sampling, an importance-sampling-based scaling technique proposed by Verdun et al. (2025), and to relative entropy coding, obtaining finite-sample policy-divergence certificates and moment-free bounds on coding cost.

Abstract: We introduce a new framework for one-step generative modelling on finite state spaces. To extend drifting beyond continuous domains, we use discrete Wasserstein geometry to define a target-relative KL gradient flow over the transitions of a reversible Markov kernel. We realize this probability flow at the particle level through Markov jumps and amortize the resulting transport updates into a latent-conditioned generator, so that the iterative dynamics are required only during training while inference remains one-step. In a controlled setting where the underlying distributions and transport dynamics can be computed exactly, we verify KL dissipation, consistency between the particle dynamics and the probability flow, and the predicted numerical scaling. We further show that a finite-capacity neural generator can track these exact transport targets while retaining one-step generation. These results validate the basic construction and provide a foundation for scaling Discrete Drifting to structured discrete data.

Abstract: An oblivious subspace embedding (OSE) is a distribution over matrices that approximately preserves the squared Euclidean norm of every vector in any fixed low-dimensional subspace. We prove the Nelson-Nguyen conjecture: for every $0 < δ< 1$, there exists a distribution that gives an OSE with embedding dimension $O((d + \log(1/δ))/\varepsilon^2)$ and column sparsity $s = O(\log(d/δ)/\varepsilon)$, with failure probability at most $δ$. We first bound the mean spectral error using a trace-moment argument and then upgrade this bound to the desired high-probability guarantee using concentration and resampling. ChatGPT-5.6-Pro was used in proving and writing the results of this manuscript.