data compression developers

Abstract: For fixed nonempty words $u,v$, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree $r$, the minimum over masks with commuting source projections is exactly $r-1$; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.

Abstract: Random composition trees model the hierarchical semantic structures of meaningful texts as a statistical ensemble generated by recursively splitting texts into coherent chunks. A recent study shows that different text types can be described by random composition trees with distinct branching factors, and that the entropy rates of these trees exhibit a similar dependence across text types as the entropy rates of the actual texts. Motivated by this correspondence, we analyze information-theoretic properties of this stochastic splitting model. First, we demonstrate that the entropy of random composition trees grows linearly with the root size, and we derive an analytical expression for the entropy rate as a function of the branching factor. Second, we prove that large trees exhibit typicality: with high probability, they belong to a typical set in which each tree is approximately equally likely. Our analysis reveals that the random composition tree model shares two key information-theoretic properties of language when modeled as a stationary and ergodic stochastic process: linear entropy growth and typicality.

Abstract: A transformer trained on sequences whose source parameters are drawn afresh each time can learn only what the sequences share: the source family and its prior. The Bayesian predictor that knows both, which we call informed Bayes, is the unique minimizer of the training loss and is computable from the family alone; we use it as a quantitative, falsifiable reference. Where earlier work compared transformers with Bayes on sources whose contexts can be learned independently, we build Markov sources with up to 64 symbols in which observations in one context inform prediction in another: product sources with known or hidden encodings, transition rows sharing an unknown base law and depth, and a mixture of three families. Our main result is that informed Bayes predicts, from the family and prior alone, how a trained transformer behaves in four tests that go beyond the learning curve: the loss that remains under a source outside the training family (predicted 2.237 bits, measured 2.309), the doubling of the extra loss when two independent components change rather than one (0.345 predicted, 0.350 measured), the benefit of other contexts when a token appears for the first time, and the depth at which copying an earlier continuation starts to pay. On the learning curves themselves, small transformers match informed Bayes when the shared structure is visible in the tokens, and hidden structure needs more layers and training, with gaps below 0.01 bits per token for the largest models. We explain why the imitation is possible and where it must fail: informed Bayes decomposes into sufficient statistics that attention selects and accumulates and an estimator, fixed by the prior, that the feed-forward network learns; training pays the complexity of informed Bayes under the architecture's prior, and finite-memory universal prediction bounds how closely a bounded network can reach it.

Abstract: Solomonoff induction is a general solution for sequence prediction that started the field of algorithmic information theory around 1960. Let $M$ be the universal semimeasure. After observing a string $ω_1 \cdots ω_{n-1}$, it predicts the next bit $ω_n$ to be $b$ with conditional probability proportional to $M(ω_1 \cdots ω_{n-1} b)$. Thus the error of this probability is \[ e_ω(n) = 1 - \frac {M(ω_1 \cdots ω_{n-1}ω_n)} {M(ω_1 \cdots ω_{n-1} 0) + M(ω_{1} \cdots ω_{n-1} 1)}. \] If $ω$ a computable infinite sequence with a self-delimiting description of at most $k$ bits, then $\sum_n e_ω(n) \le k\ln 2+O(1)$. Lattimore, Hutter and Gavane consider the problem of predicting computable patterns in a sequence, for example, the pattern of repeated bits $ω= 00\,11\,00\,00\,11\,00\cdots$ They show that the error probabilities of the bits fixed by a computable pattern have limit zero, for example, for repeated bits, one has $\lim_n e_ω(2n) = 0$. We show that for some computable pattern, the sum of $e_ω(n)$ over positions $n$ from the pattern is not bounded. However, the sum of squares $(e_ω(n))^2$ is bounded exponentially in the program length of the pattern. It is also shown that bits of $ω$ that are fixed by a computable classification function $f:\{0,1\}^* \rightarrow \{0,1\}$, converges faster: the sum of $(e_ω(n))^2$ is bounded by a polynomial in the program length for programs whose runtime is bounded by any fixed computable time bound.

Abstract: The idea that learning is closely tied to compression has long served as both a powerful principle and, at times, an informal analogy across machine learning and learning theory. Here we focus on settings in which that intuition can be made mathematically precise, surveying formal connections among learning algorithms, data compression, and computational complexity theory, with an emphasis on recent progress and current frontiers. We describe how techniques from complexity theory can guide the design of both supervised and unsupervised learning algorithms with provable guarantees, and conversely, how learning algorithms can yield new insights and results in complexity theory. A central theme is the role of data compression as a bridge between learning and complexity, especially through an area called meta-complexity, which studies the computational difficulty of compression problems and time-bounded variants of Kolmogorov complexity.

Abstract: In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all $α$-$z$ relative entropies for $α<1$. This demonstrates the first operational interpretation of these relative entropies.

Abstract: Learned representations can lose latent degrees of freedom successively, suggesting a cascade of transitions whose underlying stability principle remains unclear. Here we formulate dimension-wise posterior collapse in variational autoencoder (VAE) as a fluctuation theory around partially collapsed states. Interpreting the negative evidence lower bound as an effective free energy, its quadratic expansion defines a Gaussian theory whose Hessian acts as a mass matrix for latent fluctuations. We show that the collapsed directions form an invariant fluctuation sector and derive its exact mass spectrum in terms of a conditional residual operator. A local reactivation direction lowers the free energy when the decoder variance falls below the residual spectral upper edge, with equality marking marginality. The criterion recovers principal component thresholds in the linear Gaussian VAE limit. Viewed in reverse along continuously connected branches, the reactivation boundary provides a local criterion for successive collapse. Numerical continuation experiments show successive loss of latent dimensions near these spectral marginalities. These results support a spectral cascade interpretation governed by residual information left unexplained by the surviving representation.

Abstract: We present the Prior-Fitted Language Model (PFLM), a 300M-parameter byte-level transformer pretrained only on samples from a synthetic non-linguistic prior. Given a prefix of real text, it learns to predict the language in context with frozen weights, having never seen a word of any real language. Every training sequence is generated by a recurrent structural causal model drawn fresh from a distribution over such models. The model never sees the same language twice during training, so the only way to predict the continuation is to infer the language from the prefix. Samples from this prior share the statistical signatures of natural text: Zipfian frequencies, slow entropy-rate convergence, and long-range dependence. On Wikipedia in six languages, bits per byte fall from the uniform eight to between 0.9 and 2.4 at one million bytes of context. Given numerals instead of text, PFLM learns to count, to compare magnitudes, and to add approximately. It predicts deterministic sequences like Rudin-Shapiro or the prime indicator, and it compresses six non-text domains, from source code to speech, below gzip and PPMd. The model has not learned a language. It has learned to learn one.

Abstract: The classical information bottleneck (IB) measures the relevance of a representation $U$ of $X$ to a target $Y$ by $I(U;Y)$, which does not directly characterize the error of downstream decisions. For a binary hypothesis $Y$ inferred from many separately encoded observations, the optimal error exponent is the Chernoff information between the two conditional distributions of $U$ given $Y$. We study the mutual information constrained Chernoff bottleneck, which seeks an encoder that maximizes this Chernoff information subject to a rate constraint $I(U;X) \leq R$. We show that its optimal value $C(R)$ increases strictly up to $R = H(V)$, where $V$ merges the symbols of $X$ with equal likelihood ratio, remains at the uncompressed exponent beyond, and, unlike the IB curve, need not be concave. We further show that $k+1$ outputs suffice to attain $C(R)$, where $k$ is the cardinality of $V$. We propose an alternating algorithm that updates the encoder via a generalized Blahut--Arimoto algorithm and the Chernoff parameter $s$ via a nonlinear equation, and prove that its iterates remain feasible, with nondecreasing and convergent Chernoff information. Numerical experiments confirm the theory, and on real topic-detection data from the 20 Newsgroups corpus, compressing each word to only $17\%$ of its entropy retains $90\%$ of the error exponent and nearly the accuracy of the uncompressed classifier.

Abstract: Whereas previous results on abilities and limitations of transformers have restricted the definition of transformers in various ways, here we study softmax-attention, multi-layer transformers operating on real values, with very few additional assumptions. Applying results from real geometry, we obtain upper bounds on the VC dimension and split VC dimension of such transformers ($O(n^4)$ and $O(n^6)$, respectively, where $n$ is the input length). Conversely, we also construct specific transformers witnessing lower bounds on these quantities ($Ω(n)$ in each case). These results have some notable consequences. For example, within the class of symmetric (permutation-invariant) functions, we show that transformers can uniformly express all functions over an alphabet of one symbol and non-uniformly express all functions over an alphabet of two symbols, but cannot (even non-uniformly) express some functions over an alphabet of six symbols. We also prove limitations on how many bits of a real number a transformer can access.

Abstract: The capacity $C(d)$ of the i.i.d. binary deletion channel exists by Dobrushin's information-stability theorem, but no closed form is known. Classical constructive lower bounds from i.i.d. run-length coding have been evaluated only for one- or two-parameter families (geometric, Markov, or Morse-type). We show that the same infinite-blocklength functionals become strictly stronger when the run-length law $P$ is treated as a free distribution and optimized by learning. We reduce the Drinea--Mitzenmacher functional to a bilinear form in $P$ and prove that finite-support truncation is one-sided, so computed values remain valid lower bounds. We extend the Venkataramanan et al. reductions from geometric runs to arbitrary finite-support laws, including a residual-run HMM for output-bit entropy. Softmax gradient ascent searches $P$; every reported number is a fresh one-sided evaluation of the formula, with no Monte Carlo and no finite length-entropy penalty. The envelope of the two optimized bounds exceeds Gallager's $1-h(d)$ (for $d<1/2$) and the tabulated bounds of Drinea--Mitzenmacher, Venkataramanan et al., and Rubinstein--Con at every tested $d$. Representative values: $C(d)\ge 0.92212$, $0.72939$, $0.56486$, $0.35127$, $0.22616$, $0.10414$, $0.02891$, $0.01322$ at $d=0.01$, $0.05$, $0.10$, $0.20$, $0.30$, $0.50$, $0.80$, $0.90$. The largest absolute gain over that record is $3.7\times 10^{-3}$ bits (at $d=0.30$); the largest relative gain is $6.8\%$ (at $d=0.90$). For $d\le 0.45$ the envelope is the free-$P$ Venkataramanan functional; from $d=0.50$ it is the learned Drinea--Mitzenmacher law. At large $d$ the optimizer finds sparse run-length combs that parametric families cannot represent. A concurrent enclosure of Papailiopoulos is stronger on much of $[0,1]$, but our envelope remains larger at high $d$ (e.g. $0.02891$ vs $0.02884$ at $d=0.80$; $0.01322$ vs $0.01293$ at $d=0.90$).