The two-parameter Mittag-Leffler function $E_{\alpha, \beta}$ is of fundamental importance in fractional calculus. It appears frequently in the solutions of fractional differential and integral equations. Nonetheless, this vital function is often expensive to compute. Several attempts have been made to construct cost-effective and accurate approximations. These attempts focus mainly on the completely monotone Mittag-Leffler functions. However, when $\alpha > 1$ the monotonicity property is largely lost and as such roots and oscillations are exhibited. Consequently, existing approximants constructed mainly for $\alpha \in (0,1)$ often fail to capture this oscillatory behavior. In this paper, we construct computationally efficient and accurate rational approximants for $E_{\alpha, \beta}(-t)$, $t \ge 0$, with $\alpha \in (1,2)$. This construction is fundamentally based on the decomposition of Mittag-Leffler function with real roots into one without and a polynomial. Following which new approximants are constructed by combining the global Pad\'e approximation with a polynomial of appropriate degree. The rational approximants are extended to approximation of matrix Mittag-Leffler and different approaches to achieve efficient implementation for matrix arguments are discussed. Numerical experiments are provided to illustrate the significant accuracy improvement achieved by the proposed approximants.
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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.
Predictive multiplicity refers to the phenomenon in which classification tasks may admit multiple competing models that achieve almost-equally-optimal performance, yet generate conflicting outputs for individual samples. This presents significant concerns, as it can potentially result in systemic exclusion, inexplicable discrimination, and unfairness in practical applications. Measuring and mitigating predictive multiplicity, however, is computationally challenging due to the need to explore all such almost-equally-optimal models, known as the Rashomon set, in potentially huge hypothesis spaces. To address this challenge, we propose a novel framework that utilizes dropout techniques for exploring models in the Rashomon set. We provide rigorous theoretical derivations to connect the dropout parameters to properties of the Rashomon set, and empirically evaluate our framework through extensive experimentation. Numerical results show that our technique consistently outperforms baselines in terms of the effectiveness of predictive multiplicity metric estimation, with runtime speedup up to $20\times \sim 5000\times$. With efficient Rashomon set exploration and metric estimation, mitigation of predictive multiplicity is then achieved through dropout ensemble and model selection.
Number theoretic transform (NTT) has been a very useful tool in computations for number theory, algebra and cryptography. Its performance affects some post-quantum cryptosystems. In this paper, we discuss the butterfly operation of NTT. This basic module of NTT requires heavy modular arithmetics. Montgomery reduction is commonly used in this setting. Recently several variants of Montgomery algorithm have been proposed for the purpose of speeding up NTT. We observe that the Chinese remainder theorem (CRT) can be involved in this type of algorithms in nature and transparent ways. In this paper, a framework of using CRT to model Montgomery type algorithms is described. The derivation of these algorithms as well as their correctness are all treated in the CRT framework. Under our approach, some problems of a modular reduction algorithm (published in IACR Transactions on Cryptographic Hardware and Embedded Systems, doi:10.46586/tches.v2022.i4.614-636 ) are identified, and a counterexample is generated to show that the algorithm is incorrect.
I study the problem of learning a Lipschitz function with corrupted binary signals. The learner tries to learn a $L$-Lipschitz function $f: [0,1]^d \rightarrow [0, L]$ that the adversary chooses. There is a total of $T$ rounds. In each round $t$, the adversary selects a context vector $x_t$ in the input space, and the learner makes a guess to the true function value $f(x_t)$ and receives a binary signal indicating whether the guess is high or low. In a total of $C$ rounds, the signal may be corrupted, though the value of $C$ is \emph{unknown} to the learner. The learner's goal is to incur a small cumulative loss. This work introduces the new algorithmic technique \emph{agnostic checking} as well as new analysis techniques. I design algorithms which: for the symmetric loss, the learner achieves regret $L\cdot O(C\log T)$ with $d = 1$ and $L\cdot O_d(C\log T + T^{(d-1)/d})$ with $d > 1$; for the pricing loss, the learner achieves regret $L\cdot \widetilde{O} (T^{d/(d+1)} + C\cdot T^{1/(d+1)})$.
Machine unlearning has emerged as a new paradigm to deliberately forget data samples from a given model in order to adhere to stringent regulations. However, existing machine unlearning methods have been primarily focused on classification models, leaving the landscape of unlearning for generative models relatively unexplored. This paper serves as a bridge, addressing the gap by providing a unifying framework of machine unlearning for image-to-image generative models. Within this framework, we propose a computationally-efficient algorithm, underpinned by rigorous theoretical analysis, that demonstrates negligible performance degradation on the retain samples, while effectively removing the information from the forget samples. Empirical studies on two large-scale datasets, ImageNet-1K and Places-365, further show that our algorithm does not rely on the availability of the retain samples, which further complies with data retention policy. To our best knowledge, this work is the first that represents systemic, theoretical, empirical explorations of machine unlearning specifically tailored for image-to-image generative models. Our code is available at //github.com/jpmorganchase/l2l-generator-unlearning.
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.
We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we provide two examples of classifier comparisons on samples of standard benchmark data sets. Our results demonstrate promisingly the wide variety of different analysis approaches based on ufg methods. Furthermore, the examples outline that our approach differs substantially from existing benchmarking approaches, and thus adds a new perspective to the vivid debate on classifier comparison.
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 $\mathsf{GDiLL}$.
This work considers an asynchronous $\textsf{K}_\text{a}$-active-user unsourced multiple access channel (AUMAC) with the worst-case asynchronicity. The transmitted messages must be decoded within $n$ channel uses, while some codewords are not completely received due to asynchronicities. We consider a constraint of the largest allowed delay of the transmission. The AUMAC lacks the permutation-invariant property of the synchronous UMAC since different permutations of the same codewords with a fixed asynchronicity are distinguishable. Hence, the analyses require calculating all $2^{\textsf{K}_\text{a}}-1$ combinations of erroneously decoded messages. Moreover, transmitters cannot adapt the corresponding codebooks according to asynchronicity due to a lack of information on asynchronicities. To overcome this challenge, a uniform bound of the per-user probability of error (PUPE) is derived by investigating the worst-case of the asynchronous patterns with the delay constraint. Numerical results show the trade-off between the energy-per-bit and the number of active users for different delay constraints. In addition, although the asynchronous transmission reduces interference, the required energy-per-bit increases as the receiver decodes with incompletely received codewords, compared to the synchronous case.
Deep Learning (DL) is vulnerable to out-of-distribution and adversarial examples resulting in incorrect outputs. To make DL more robust, several posthoc anomaly detection techniques to detect (and discard) these anomalous samples have been proposed in the recent past. This survey tries to provide a structured and comprehensive overview of the research on anomaly detection for DL based applications. We provide a taxonomy for existing techniques based on their underlying assumptions and adopted approaches. We discuss various techniques in each of the categories and provide the relative strengths and weaknesses of the approaches. Our goal in this survey is to provide an easier yet better understanding of the techniques belonging to different categories in which research has been done on this topic. Finally, we highlight the unsolved research challenges while applying anomaly detection techniques in DL systems and present some high-impact future research directions.
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