We show that marginals of blocks of $t$ systems of any finitely correlated translation invariant state on a chain can be learned, in trace distance, with $O(t^2)$ copies -- with an explicit dependence on local dimension, memory dimension and spectral properties of a certain map constructed from the state -- and computational complexity polynomial in $t$. The algorithm requires only the estimation of a marginal of a controlled size, in the worst case bounded by the minimum bond dimension, from which it reconstructs a translation invariant matrix product operator. In the analysis, a central role is played by the theory of operator systems. A refined error bound can be proven for $C^*$-finitely correlated states, which have an operational interpretation in terms of sequential quantum channels applied to the memory system. We can also obtain an analogous error bound for a class of matrix product density operators reconstructible by local marginals. In this case, a linear number of marginals must be estimated, obtaining a sample complexity of $\tilde{O}(t^3)$. The learning algorithm also works for states that are only close to a finitely correlated state, with the potential of providing competitive algorithms for other interesting families of states.
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In combinatorics on words, the well-studied factor complexity function $\rho_{\bf x}$ of a sequence ${\bf x}$ over a finite alphabet counts, for any nonnegative integer $n$, the number of distinct length-$n$ factors of ${\bf x}$. In this paper, we introduce the \emph{reflection complexity} function $r_{\bf x}$ to enumerate the factors occurring in a sequence ${\bf x}$, up to reversing the order of symbols in a word. We introduce and prove results on $r_{\bf x}$ regarding its growth properties and relationship with other complexity functions. We prove that if ${\bf x}$ is $k$-automatic, then $r_{\bf x}$ is computably $k$-regular, and we use the software {\tt Walnut} to evaluate the reflection complexity of automatic sequences, such as the Thue--Morse sequence. We prove a Morse--Hedlund-type result characterizing eventually periodic sequences in terms of their reflection complexity, and we deduce a characterization of Sturmian sequences. Furthermore, we investigate the reflection complexity of episturmian, $(s+1)$-dimensional billiard, and Rote sequences. There are still many unanswered questions about this measure.
A tiling of a vector space $S$ is the pair $(U,V)$ of its subsets such that every vector in $S$ is uniquely represented as the sum of a vector from $U$ and a vector from $V$. A tiling is connected to a perfect codes if one of the sets, say $U$, is projective, i.e., the union of one-dimensional subspaces of $S$. A tiling $(U,V)$ is full-rank if the affine span of each of $U$, $V$ is $S$. For finite non-binary vector spaces of dimension at least $6$ (at least $10$), we construct full-rank tilings $(U,V)$ with projective $U$ (both $U$ and $V$, respectively). In particular, that construction gives a full-rank ternary $1$-perfect code of length $13$, solving a known problem. We also discuss the treatment of tilings with projective components as factorizations of projective spaces. Keywords: perfect codes, tilings, group factorization, full-rank tilings, projective geometry
In probabilistic modelling, joint distributions are often of more interest than their marginals, but the standard composition of stochastic channels is defined by marginalization. Recently, the notion of 'copy-composition' was introduced in order to circumvent this problem and express the chain rule of the relative entropy fibrationally, but while that goal was achieved, copy-composition lacked a satisfactory origin story. Here, we supply such a story for two standard probabilistic tools: directed and undirected graphical models. We explain that (directed) Bayesian networks may be understood as "stochastic terms" of product type, in which context copy-composition amounts to a pull-push operation. Likewise, we show that (undirected) factor graphs compose by copy-composition. In each case, our construction yields a double fibration of decorated (co)spans. Along the way, we introduce a useful bifibration of measure kernels, to provide semantics for the notion of stochastic term, which allows us to generalize probabilistic modelling from product to dependent types.
Nonparametric estimators for the mean and the covariance functions of functional data are proposed. The setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The estimators depend on the local regularity of the stochastic process generating the functional data. We consider a simple estimator of this local regularity which exploits the replication and regularization features of functional data. Next, we use the ``smoothing first, then estimate'' approach for the mean and the covariance functions. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon an example of real data set, illustrate the effectiveness of the new approach.
Retrieval augmented generation (RAG) is frequently used to mitigate hallucinations and provide up-to-date knowledge for large language models (LLMs). However, given that document retrieval is an imprecise task and sometimes results in erroneous or even harmful content being presented in context, this raises the question of how LLMs handle retrieved information: If the provided content is incorrect, does the model know to ignore it, or does it recapitulate the error? Conversely, when the model's initial response is incorrect, does it always know to use the retrieved information to correct itself, or does it insist on its wrong prior response? To answer this, we curate a dataset of over 1200 questions across six domains (e.g., drug dosages, Olympic records, locations) along with content relevant to answering each question. We further apply precise perturbations to the answers in the content that range from subtle to blatant errors. We benchmark six top-performing LLMs, including GPT-4o, on this dataset and find that LLMs are susceptible to adopting incorrect retrieved content, overriding their own correct prior knowledge over 60% of the time. However, the more unrealistic the retrieved content is (i.e. more deviated from truth), the less likely the model is to adopt it. Also, the less confident a model is in its initial response (via measuring token probabilities), the more likely it is to adopt the information in the retrieved content. We exploit this finding and demonstrate simple methods for improving model accuracy where there is conflicting retrieved content. Our results highlight a difficult task and benchmark for LLMs -- namely, their ability to correctly discern when it is wrong in light of correct retrieved content and to reject cases when the provided content is incorrect.
Reliably measuring the collinearity of bivariate data is crucial in statistics, particularly for time-series analysis or ongoing studies in which incoming observations can significantly impact current collinearity estimates. Leveraging identities from Welford's online algorithm for sample variance, we develop a rigorous theoretical framework for analyzing the maximal change to the Pearson correlation coefficient and its p-value that can be induced by additional data. Further, we show that the resulting optimization problems yield elegant closed-form solutions that can be accurately computed by linear- and constant-time algorithms. Our work not only creates new theoretical avenues for robust correlation measures, but also has broad practical implications for disciplines that span econometrics, operations research, clinical trials, climatology, differential privacy, and bioinformatics. Software implementations of our algorithms in Cython-wrapped C are made available at //github.com/marc-harary/sensitivity for reproducibility, practical deployment, and future theoretical development.
We set up, at the abstract Hilbert space setting, the general question on when an inverse linear problem induced by an operator of Friedrichs type admits solutions belonging to (the closure of) the Krylov subspace associated to such operator. Such Krylov solvability of abstract Friedrichs systems allows to predict when, for concrete differential inverse problems, truncation algorithms can or cannot reproduce the exact solutions in terms of approximants from the Krylov subspace.
Intelligent autonomous systems are part of a system of systems that interact with other agents to accomplish tasks in complex environments. However, intelligent autonomous systems integrated system of systems add additional layers of complexity based on their limited cognitive processes, specifically shared situation awareness that allows a team to respond to novel tasks. Intelligent autonomous systems' lack of shared situation awareness adversely influences team effectiveness in complex task environments, such as military command-and-control. A complementary approach of shared situation awareness, called situations theory, is beneficial for understanding the relationship between system of systems shared situation awareness and effectiveness. The current study elucidates a conceptual discussion on situations theory to investigate the development of an system of systems shared situational awareness when humans team with intelligent autonomous system agents. To ground the discussion, the reviewed studies expanded situations theory within the context of a system of systems that result in three major conjectures that can be beneficial to the design and development of future systems of systems.
Suppose that $P$ is a property that may be satisfied by a random code $C \subset \Sigma^n$. For example, for some $p \in (0,1)$, ${P}$ might be the property that there exist three elements of $C$ that lie in some Hamming ball of radius $pn$. We say that $R^*$ is the threshold rate for ${P}$ if a random code of rate $R^* + \epsilon$ is very likely to satisfy ${P}$, while a random code of rate $R^* - \epsilon$ is very unlikely to satisfy ${P}$. While random codes are well-studied in coding theory, even the threshold rates for relatively simple properties like the one above are not well understood. We characterize threshold rates for a rich class of properties. These properties, like the example above, are defined by the inclusion of specific sets of codewords which are also suitably "symmetric". For properties in this class, we show that the threshold rate is in fact equal to the lower bound that a simple first-moment calculation obtains. Our techniques not only pin down the threshold rate for the property ${P}$ above, they give sharp bounds on the threshold rate for list-recovery in several parameter regimes, as well as an efficient algorithm for estimating the threshold rates for list-recovery in general.
The associahedron is the graph $\mathcal{G}_N$ that has as nodes all triangulations of a convex $N$-gon, and an edge between any two triangulations that differ in a flip operation, which consists of removing an edge shared by two triangles and replacing it by the other diagonal of the resulting 4-gon. In this paper, we consider a large collection of induced subgraphs of $\mathcal{G}_N$ obtained by Ramsey-type colorability properties. Specifically, coloring the points of the $N$-gon red and blue alternatingly, we consider only colorful triangulations, namely triangulations in which every triangle has points in both colors, i.e., monochromatic triangles are forbidden. The resulting induced subgraph of $\mathcal{G}_N$ on colorful triangulations is denoted by $\mathcal{F}_N$. We prove that $\mathcal{F}_N$ has a Hamilton cycle for all $N\geq 8$, resolving a problem raised by Sagan, i.e., all colorful triangulations on $N$ points can be listed so that any two cyclically consecutive triangulations differ in a flip. In fact, we prove that for an arbitrary fixed coloring pattern of the $N$ points with at least 10 changes of color, the resulting subgraph of $\mathcal{G}_N$ on colorful triangulations (for that coloring pattern) admits a Hamilton cycle. We also provide an efficient algorithm for computing a Hamilton path in $\mathcal{F}_N$ that runs in time $\mathcal{O}(1)$ on average per generated node. This algorithm is based on a new and algorithmic construction of a tree rotation Gray code for listing all $n$-vertex $k$-ary trees that runs in time $\mathcal{O}(k)$ on average per generated tree.
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