We consider the problem of learning a graph modeling the statistical relations of the $d$ variables from a dataset with $n$ samples $X \in \mathbb{R}^{n \times d}$. Standard approaches amount to searching for a precision matrix $\Theta$ representative of a Gaussian graphical model that adequately explains the data. However, most maximum likelihood-based estimators usually require storing the $d^{2}$ values of the empirical covariance matrix, which can become prohibitive in a high-dimensional setting. In this work, we adopt a compressive viewpoint and aim to estimate a sparse $\Theta$ from a \emph{sketch} of the data, i.e. a low-dimensional vector of size $m \ll d^{2}$ carefully designed from $X$ using non-linear random features. Under certain assumptions on the spectrum of $\Theta$ (or its condition number), we show that it is possible to estimate it from a sketch of size $m=\Omega\left((d+2k)\log(d)\right)$ where $k$ is the maximal number of edges of the underlying graph. These information-theoretic guarantees are inspired by compressed sensing theory and involve restricted isometry properties and instance optimal decoders. We investigate the possibility of achieving practical recovery with an iterative algorithm based on the graphical lasso, viewed as a specific denoiser. We compare our approach and graphical lasso on synthetic datasets, demonstrating its favorable performance even when the dataset is compressed.
相關內容
Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to \{0,1\}^k$ is said to be a direct product function if there exists a function $f\colon X(1)\to \{0,1\}$ such that $F(\sigma) = (f(\sigma_1), \ldots, f(\sigma_k))$ for each $k$-face $\sigma$. In an effort to simplify components of the PCP theorem, Goldreich and Safra introduced the problem of direct product testing, which asks whether one can test if $F\colon X(k)\to \{0,1\}^k$ is correlated with a direct product function by querying $F$ on only $2$ inputs. Dinur and Kaufman conjectured that there exist bounded degree complexes with a direct product test in the small soundness regime. We resolve their conjecture by showing that for all $\delta>0$, there exists a family of high-dimensional expanders with degree $O_{\delta}(1)$ and a $2$-query direct product tester with soundness $\delta$. We use the characterization given by a subset of the authors and independently by Dikstein and Dinur, who showed that some form of non-Abelian coboundary expansion (which they called "Unique-Games coboundary expansion") is a necessary and sufficient condition for a complex to admit such direct product testers. Our main technical contribution is a general technique for showing coboundary expansion of complexes with coefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders constructed by Chapman and Lubotzky satisfies the necessary conditions, thus admitting a 2-query direct product tester with small soundness.
The automorphism groups of various linear codes are well-studied yielding valuable insights into the respective code structure. This knowledge is successfully applied in, e.g., theoretical analysis and in improving decoding performance motivating the analyses of endomorphisms of linear codes. In this work, we discuss the structure of the set of transformation matrices of code endomorphisms, defined as a generalization of code automorphisms, and provide an explicit construction of a bijective mapping between the image of an endomorphism and its canonical quotient space. Furthermore, we introduce a one-to-one mapping between the set of transformation matrices of endomorphisms and a larger linear block code enabling the use of well-known algorithms for the search for suitable endomorphisms. Additionally, we propose an approach to obtain unknown code endomorphisms based on automorphisms of the code. Furthermore, we consider ensemble decoding as a possible use case for endomorphisms by introducing endomorphism ensemble decoding. Interestingly, EED can improve decoding performance when other ensemble decoding schemes are not applicable.
A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA, for short). We prove a representability theorem: for each NFA $N$, there exists a process algebraic term $p$ such that its semantics is an NFA isomorphic to $N$. Moreover, we provide a concise axiomatization of language equivalence: two NFAs $N_1$ and $N_2$ recognize the same language if and only if the associated terms $p_1$ and $p_2$, respectively, can be equated by means of a set of axioms, comprising 7 axioms plus 3 conditional axioms, only.
We give an example of a class of distributions that is learnable in total variation distance with a finite number of samples, but not learnable under $(\varepsilon, \delta)$-differential privacy. This refutes a conjecture of Ashtiani.
In Linear Logic ($\mathsf{LL}$), the exponential modality $!$ brings forth a distinction between non-linear proofs and linear proofs, where linear means using an argument exactly once. Differential Linear Logic ($\mathsf{DiLL}$) is an extension of Linear Logic which includes additional rules for $!$ which encode differentiation and the ability of linearizing proofs. On the other hand, Graded Linear Logic ($\mathsf{GLL}$) is a variation of Linear Logic in such a way that $!$ is now indexed over a semiring $R$. This $R$-grading allows for non-linear proofs of degree $r \in R$, such that the linear proofs are of degree $1 \in R$. There has been recent interest in combining these two variations of $\mathsf{LL}$ together and developing Graded Differential Linear Logic ($\mathsf{GDiLL}$). In this paper we present a sequent calculus for $\mathsf{GDiLL}$, as well as introduce its categorical semantics, which we call graded differential categories, using both coderelictions and deriving transformations. We prove that symmetric powers always give graded differential categories, and provide other examples of graded differential categories. We also discuss graded versions of (monoidal) coalgebra modalities, additive bialgebra modalities, and the Seely isomorphisms, as well as their implementations in the sequent calculus of $\mathsf{GDiLL}$.
We consider a boundary value problem (BVP) modelling one-dimensional heat-conduction with radiation, which is derived from the Stefan-Boltzmann law. The problem strongly depends on the parameters, making difficult to estimate the solution. We use an analytical approach to determine upper and lower bounds to the exact solution of the BVP, which allows estimating the latter. Finally, we support our theoretical arguments with numerical data, by implementing them into the MAPLE computer program.
The laws of model size, data volume, computation and model performance have been extensively studied in the field of Natural Language Processing (NLP). However, the scaling laws in Optical Character Recognition (OCR) have not yet been investigated. To address this, we conducted comprehensive studies that involved examining the correlation between performance and the scale of models, data volume and computation in the field of text recognition.Conclusively, the study demonstrates smooth power laws between performance and model size, as well as training data volume, when other influencing factors are held constant. Additionally, we have constructed a large-scale dataset called REBU-Syn, which comprises 6 million real samples and 18 million synthetic samples. Based on our scaling law and new dataset, we have successfully trained a scene text recognition model, achieving a new state-ofthe-art on 6 common test benchmarks with a top-1 average accuracy of 97.42%. The models and dataset are publicly available at //github.com/large-ocr-model/large-ocr-model.github.io.
We present a theoretical analysis of the performance of transformer with softmax attention in in-context learning with linear regression tasks. While the existing literature predominantly focuses on the convergence of transformers with single-/multi-head attention, our research centers on comparing their performance. We conduct an exact theoretical analysis to demonstrate that multi-head attention with a substantial embedding dimension performs better than single-head attention. When the number of in-context examples D increases, the prediction loss using single-/multi-head attention is in O(1/D), and the one for multi-head attention has a smaller multiplicative constant. In addition to the simplest data distribution setting, we consider more scenarios, e.g., noisy labels, local examples, correlated features, and prior knowledge. We observe that, in general, multi-head attention is preferred over single-head attention. Our results verify the effectiveness of the design of multi-head attention in the transformer architecture.
Tailoring polar code construction for decoding algorithms beyond successive cancellation has remained a topic of significant interest in the field. However, despite the inherent nested structure of polar codes, the use of sequence models in polar code construction is understudied. In this work, we propose using a sequence modeling framework to iteratively construct a polar code for any given length and rate under various channel conditions. Simulations show that polar codes designed via sequential modeling using transformers outperform both 5G-NR sequence and Density Evolution based approaches for both AWGN and Rayleigh fading channels.
Neural machine translation (NMT) is a deep learning based approach for machine translation, which yields the state-of-the-art translation performance in scenarios where large-scale parallel corpora are available. Although the high-quality and domain-specific translation is crucial in the real world, domain-specific corpora are usually scarce or nonexistent, and thus vanilla NMT performs poorly in such scenarios. Domain adaptation that leverages both out-of-domain parallel corpora as well as monolingual corpora for in-domain translation, is very important for domain-specific translation. In this paper, we give a comprehensive survey of the state-of-the-art domain adaptation techniques for NMT.
小貼士
登錄享
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191