A sparse polynomial (also called a lacunary polynomial) is a polynomial that has relatively few terms compared to its degree. The sparse-representation of a polynomial represents the polynomial as a list of its non-zero terms (coefficient-degree pairs). In particular, the degree of a sparse polynomial can be exponential in the sparse-representation size. We prove that for monic polynomials $f, g \in \mathbb{C}[x]$ such that $g$ divides $f$, the $\ell_2$-norm of the quotient polynomial $f/g$ is bounded by $\lVert f \rVert_1 \cdot \tilde{O}(\lVert{g}\rVert_0^3\text{deg}^2{ f})^{\lVert{g}\rVert_0 - 1}$. This improves upon the exponential (in $\text{deg}{ f}$) bounds for general polynomials and implies that the trivial long division algorithm runs in time quasi-linear in the input size and number of terms of the quotient polynomial $f/g$, thus solving a long-standing problem on exact divisibility of sparse polynomials. We also study the problem of bounding the number of terms of $f/g$ in some special cases. When $f, g \in \mathbb{Z}[x]$ and $g$ is a cyclotomic-free (i.e., it has no cyclotomic factors) trinomial, we prove that $\lVert{f/g}\rVert_0 \leq O(\lVert{f}\rVert_0 \text{size}({f})^2 \cdot \log^6{\text{deg}{ g}})$. When $g$ is a binomial with $g(\pm 1) \neq 0$, we prove that the sparsity is at most $O(\lVert{f}\rVert_0 ( \log{\lVert{f}\rVert_0} + \log{\lVert{f}\rVert_{\infty}}))$. Both upper bounds are polynomial in the input-size. We leverage these results and give a polynomial time algorithm for deciding whether a cyclotomic-free trinomial divides a sparse polynomial over the integers. As our last result, we present a polynomial time algorithm for testing divisibility by pentanomials over small finite fields when $\text{deg}{ f} = \tilde{O}(\text{deg}{ g})$.
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Adversarial examples in machine learning has emerged as a focal point of research due to their remarkable ability to deceive models with seemingly inconspicuous input perturbations, potentially resulting in severe consequences. In this study, we embark on a comprehensive exploration of adversarial machine learning models, shedding light on their intrinsic complexity and interpretability. Our investigation reveals intriguing links between machine learning model complexity and Einstein's theory of special relativity, through the concept of entanglement. More specific, we define entanglement computationally and demonstrate that distant feature samples can exhibit strong correlations, akin to entanglement in quantum realm. This revelation challenges conventional perspectives in describing the phenomenon of adversarial transferability observed in contemporary machine learning models. By drawing parallels with the relativistic effects of time dilation and length contraction during computation, we gain deeper insights into adversarial machine learning, paving the way for more robust and interpretable models in this rapidly evolving field.
In the pooled data problem, the goal is to identify the categories associated with a large collection of items via a sequence of pooled tests. Each pooled test reveals the number of items of each category within the pool. We study an approximate message passing (AMP) algorithm for estimating the categories and rigorously characterize its performance, in both the noiseless and noisy settings. For the noiseless setting, we show that the AMP algorithm is equivalent to one recently proposed by El Alaoui et al. Our results provide a rigorous version of their performance guarantees, previously obtained via non-rigorous techniques. For the case of pooled data with two categories, known as quantitative group testing (QGT), we use the AMP guarantees to compute precise limiting values of the false positive rate and the false negative rate. Though the pooled data problem and QGT are both instances of estimation in a linear model, existing AMP theory cannot be directly applied since the design matrices are binary valued. The key technical ingredient in our result is a rigorous analysis of AMP for generalized linear models defined via generalized white noise design matrices. This result, established using a recent universality result of Wang et al., is of independent interest. Our theoretical results are validated by numerical simulations. For comparison, we propose estimators based on convex relaxation and iterative thresholding, without providing theoretical guarantees. Our simulations indicate that AMP outperforms the convex programming estimator for a range of QGT scenarios, but the convex program performs better for pooled data with three categories.
System programming languages are typically compiled in a linear pipeline process, which is a completely opaque and isolated to end-users. This limits the possibilities of performing meta-programming in the same language and environment, and the extensibility of the compiler itself by end-users. We propose a novel redefinition of the compilation process in terms of interpreting the program definition as a script. This evaluation is performed in an environment where the full compilation pipeline is implemented and exposed to the user via a meta-object protocol, which forms the basis for a meta-circular definition and implementation of the programming language itself. We demonstrate the feasibility of this approach by bootstrapping a self-compiling implementation of Sysmel, a static and dynamic typed Smalltalk and C++ inspired programming language.
We present a comprehensive evaluation of the robustness and explainability of ResNet-like models in the context of Unintended Radiated Emission (URE) classification and suggest a new approach leveraging Neural Stochastic Differential Equations (SDEs) to address identified limitations. We provide an empirical demonstration of the fragility of ResNet-like models to Gaussian noise perturbations, where the model performance deteriorates sharply and its F1-score drops to near insignificance at 0.008 with a Gaussian noise of only 0.5 standard deviation. We also highlight a concerning discrepancy where the explanations provided by ResNet-like models do not reflect the inherent periodicity in the input data, a crucial attribute in URE detection from stable devices. In response to these findings, we propose a novel application of Neural SDEs to build models for URE classification that are not only robust to noise but also provide more meaningful and intuitive explanations. Neural SDE models maintain a high F1-score of 0.93 even when exposed to Gaussian noise with a standard deviation of 0.5, demonstrating superior resilience to ResNet models. Neural SDE models successfully recover the time-invariant or periodic horizontal bands from the input data, a feature that was conspicuously missing in the explanations generated by ResNet-like models. This advancement presents a small but significant step in the development of robust and interpretable models for real-world URE applications where data is inherently noisy and assurance arguments demand interpretable machine learning predictions.
Unbalanced Job Approximation - UJA is a family of low-cost formulas to obtain the throughput of Queueing Networks - QNs with fixed rate servers using Taylor series expansion of job loadings with respect to the mean loading. UJA with one term yields the same throughput as optimistic Balanced Job Bound - BJB, which at some point exceeds the maximum asymptotic throughput. The accuracy of the estimated throughput increases with more terms in the Taylor series. UJA can be used in parametric studies by reducing the cost of solving large QNs by aggregating stations into a single Flow-Equivalent Service Center - FESCs defined by its throughput characteristic. While UJA has been extended to two classes it may be applied to more classes by job class aggregation. BJB has been extended to QNs with delay servers and multiple jobs classes by Eager and Sevcik, throughput bounds by Eager and Sevcik, Kriz, Proportional Bound - PB and Prop. Approximation Bound - PAM by Hsieh and Lam and Geometric Bound - GB by Casale et al. are reviewed.
We propose the use of a lower or upper triangular sub-base matrix to replace the identity matrix in the source-check-channel-variable linking protomatrix of a double-protograph low-density parity-check joint-source-channel code (DP-LDPC JSCC). The elements along the diagonal of the proposed lower or upper triangular sub-base matrix are assigned as "1" and the other non-zero elements can take any non-negative integral values. Compared with the traditional DP-LDPC JSCC designs, the new designs show a theoretical channel threshold improvement of up to 0.41 dB and a simulated source symbol error rate improvement of up to 0.5 dB at an error rate of 1e-6.
Traffic congestion is a persistent problem in our society. Existing methods for traffic control have proven futile in alleviating current congestion levels leading researchers to explore ideas with robot vehicles given the increased emergence of vehicles with different levels of autonomy on our roads. This gives rise to mixed traffic control, where robot vehicles regulate human-driven vehicles through reinforcement learning (RL). However, most existing studies use precise observations that involve global information, such as environment outflow, and local information, i.e., vehicle positions and velocities. Obtaining this information requires updating existing road infrastructure with vast sensor environments and communication to potentially unwilling human drivers. We consider image observations as the alternative for mixed traffic control via RL: 1) images are ubiquitous through satellite imagery, in-car camera systems, and traffic monitoring systems; 2) images do not require a complete re-imagination of the observation space from environment to environment; and 3) images only require communication to equipment. In this work, we show robot vehicles using image observations can achieve similar performance to using precise information on environments, including ring, figure eight, intersection, merge, and bottleneck. In certain scenarios, our approach even outperforms using precision observations, e.g., up to 26% increase in average vehicle velocity in the merge environment and a 6% increase in outflow in the bottleneck environment, despite only using local traffic information as opposed to global traffic information.
Human-in-the-loop aims to train an accurate prediction model with minimum cost by integrating human knowledge and experience. Humans can provide training data for machine learning applications and directly accomplish some tasks that are hard for computers in the pipeline with the help of machine-based approaches. In this paper, we survey existing works on human-in-the-loop from a data perspective and classify them into three categories with a progressive relationship: (1) the work of improving model performance from data processing, (2) the work of improving model performance through interventional model training, and (3) the design of the system independent human-in-the-loop. Using the above categorization, we summarize major approaches in the field, along with their technical strengths/ weaknesses, we have simple classification and discussion in natural language processing, computer vision, and others. Besides, we provide some open challenges and opportunities. This survey intends to provide a high-level summarization for human-in-the-loop and motivates interested readers to consider approaches for designing effective human-in-the-loop solutions.
Recently, Mutual Information (MI) has attracted attention in bounding the generalization error of Deep Neural Networks (DNNs). However, it is intractable to accurately estimate the MI in DNNs, thus most previous works have to relax the MI bound, which in turn weakens the information theoretic explanation for generalization. To address the limitation, this paper introduces a probabilistic representation of DNNs for accurately estimating the MI. Leveraging the proposed MI estimator, we validate the information theoretic explanation for generalization, and derive a tighter generalization bound than the state-of-the-art relaxations.
While it is nearly effortless for humans to quickly assess the perceptual similarity between two images, the underlying processes are thought to be quite complex. Despite this, the most widely used perceptual metrics today, such as PSNR and SSIM, are simple, shallow functions, and fail to account for many nuances of human perception. Recently, the deep learning community has found that features of the VGG network trained on the ImageNet classification task has been remarkably useful as a training loss for image synthesis. But how perceptual are these so-called "perceptual losses"? What elements are critical for their success? To answer these questions, we introduce a new Full Reference Image Quality Assessment (FR-IQA) dataset of perceptual human judgments, orders of magnitude larger than previous datasets. We systematically evaluate deep features across different architectures and tasks and compare them with classic metrics. We find that deep features outperform all previous metrics by huge margins. More surprisingly, this result is not restricted to ImageNet-trained VGG features, but holds across different deep architectures and levels of supervision (supervised, self-supervised, or even unsupervised). Our results suggest that perceptual similarity is an emergent property shared across deep visual representations.
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