Pointwise equidistribution for almost smooth functions with an error rate and Weighted L\'evy-Khintchin theorem

The purpose of this article is twofold: to prove a pointwise equidistribution theorem with an error rate for almost smooth functions, which strengthens the main result of Kleinbock, Shi and Weiss (2017); and to obtain a L\'evy-Khintchin theorem for weighted best approximations, which extends the main theorem of Cheung and Chevallier (2019). To do so, we employ techniques from homogeneous dynamics and the methods developed in the work of Cheung-Chevallier (2019) and Shapira-Weiss (2022).Comment: 32 page

Calculation of static transmission errors associated with thermo-elastic coupling contacts of spur gears

The static transmission error is one of the key incentives of gear dynamics and addressed by many scholars. However, the traditional calculation method of static transmission errors of spur gears does not take into account the influence of thermo-elastic coupling caused by the gear temperature field, and it limits the accuracy of the dynamic characteristic analysis. Thus, in this study, the calculation method of static transmission errors associated with thermo-elastic coupling is proposed. Furthermore, the differences between static transmission errors associated with thermo-elastic coupling contacts and traditional method of gear is discussed. The study is helpful to improve the accuracy of dynamic analysis of gear transmission system

Discrete forecast reconciliation

While forecast reconciliation has seen great success for real valued data, the method has not yet been comprehensively extended to the discrete case. This paper defines and develops a formal discrete forecast reconciliation framework based on optimising scoring rules using quadratic programming. The proposed framework produces coherent joint probabilistic forecasts for count hierarchical timeTwo discrete reconciliation algorithms are proposed and compared to generalisations of the top-down and bottom-up approaches to count data. Two simulation experiments and two empirical examples are conducted to validate that the proposed reconciliation algorithms improve forecast accuracy. The empirical applications are to forecast criminal offences in Washington D.C. and the exceedance of thresholds in age-specific mortality rates in Australia. Compared to the top-down and bottom-up approaches, the proposed framework shows superior performance in both simulations and empirical studies

A supramolecular radical cation: folding-enhanced electrostatic effect for promoting radical-mediated oxidation.

We report a supramolecular strategy to promote radical-mediated Fenton oxidation by the rational design of a folded host-guest complex based on cucurbit[8]uril (CB[8]). In the supramolecular complex between CB[8] and a derivative of 1,4-diketopyrrolo[3,4-c]pyrrole (DPP), the carbonyl groups of CB[8] and the DPP moiety are brought together through the formation of a folded conformation. In this way, the electrostatic effect of the carbonyl groups of CB[8] is fully applied to highly improve the reactivity of the DPP radical cation, which is the key intermediate of Fenton oxidation. As a result, the Fenton oxidation is extraordinarily accelerated by over 100 times. It is anticipated that this strategy could be applied to other radical reactions and enrich the field of supramolecular radical chemistry in radical polymerization, photocatalysis, and organic radical battery and holds potential in supramolecular catalysis and biocatalysis

Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions

We provide a simple convergence proof for AdaGrad optimizing non-convex objectives under only affine noise variance and bounded smoothness assumptions. The proof is essentially based on a novel auxiliary function $\xi$ that helps eliminate the complexity of handling the correlation between the numerator and denominator of AdaGrad's update. Leveraging simple proofs, we are able to obtain tighter results than existing results \citep{faw2022power} and extend the analysis to several new and important cases. Specifically, for the over-parameterized regime, we show that AdaGrad needs only $\mathcal{O}(\frac{1}{\varepsilon^2})$ iterations to ensure the gradient norm smaller than $\varepsilon$, which matches the rate of SGD and significantly tighter than existing rates $\mathcal{O}(\frac{1}{\varepsilon^4})$ for AdaGrad. We then discard the bounded smoothness assumption and consider a realistic assumption on smoothness called $(L_0,L_1)$-smooth condition, which allows local smoothness to grow with the gradient norm. Again based on the auxiliary function $\xi$, we prove that AdaGrad succeeds in converging under $(L_0,L_1)$-smooth condition as long as the learning rate is lower than a threshold. Interestingly, we further show that the requirement on learning rate under the $(L_0,L_1)$-smooth condition is necessary via proof by contradiction, in contrast with the case of uniform smoothness conditions where convergence is guaranteed regardless of learning rate choices. Together, our analyses broaden the understanding of AdaGrad and demonstrate the power of the new auxiliary function in the investigations of AdaGrad.Comment: COLT 202

Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity

Recently, Arjevani et al. [1] established a lower bound of iteration complexity for the first-order optimization under an $L$-smooth condition and a bounded noise variance assumption. However, a thorough review of existing literature on Adam's convergence reveals a noticeable gap: none of them meet the above lower bound. In this paper, we close the gap by deriving a new convergence guarantee of Adam, with only an $L$-smooth condition and a bounded noise variance assumption. Our results remain valid across a broad spectrum of hyperparameters. Especially with properly chosen hyperparameters, we derive an upper bound of the iteration complexity of Adam and show that it meets the lower bound for first-order optimizers. To the best of our knowledge, this is the first to establish such a tight upper bound for Adam's convergence. Our proof utilizes novel techniques to handle the entanglement between momentum and adaptive learning rate and to convert the first-order term in the Descent Lemma to the gradient norm, which may be of independent interest.Comment: NeurIPS 2023 Accep

End-to-end One-shot Human Parsing

Previous human parsing methods are limited to parsing humans into pre-defined classes, which is inflexible for practical fashion applications that often have new fashion item classes. In this paper, we define a novel one-shot human parsing (OSHP) task that requires parsing humans into an open set of classes defined by any test example. During training, only base classes are exposed, which only overlap with part of the test-time classes. To address three main challenges in OSHP, i.e., small sizes, testing bias, and similar parts, we devise an End-to-end One-shot human Parsing Network (EOP-Net). Firstly, an end-to-end human parsing framework is proposed to parse the query image into both coarse-grained and fine-grained human classes, which embeds rich semantic information that is shared across different granularities to identify the small-sized human classes. Then, we gradually smooth the training-time static prototypes to get robust class representations. Moreover, we employ a dynamic objective to encourage the network's enhancing features' representational capability in the early training phase while improving features' transferability in the late training phase. Therefore, our method can quickly adapt to the novel classes and mitigate the testing bias issue. In addition, we add a contrastive loss at the prototype level to enforce inter-class distances, thereby discriminating the similar parts. For comprehensive evaluations on the new task, we tailor three existing popular human parsing benchmarks to the OSHP task. Experiments demonstrate that EOP-Net outperforms representative one-shot segmentation models by large margins and serves as a strong baseline for further research. The source code is available at https://github.com/Charleshhy/One-shot-Human-Parsing.Comment: Accepted to IEEE Trans. Pattern Analysis and Machine Intelligence (TPAMI