We consider the optimization of a smooth and strongly convex objective using constant step-size stochastic gradient descent (SGD) and study its properties through the prism of Markov chains. We show that, for unbiased gradient estimates with mildly controlled variance, the iteration converges to an invariant distribution in total variation distance. We also establish this convergence in Wasserstein-2 distance in a more general setting compared to previous work. Thanks to the invariance property of the limit distribution, our analysis shows that the latter inherits sub-Gaussian or sub-exponential concentration properties when these hold true for the gradient. This allows the derivation of high-confidence bounds for the final estimate. Finally, under such conditions in the linear case, we obtain a dimension-free deviation bound for the Polyak-Ruppert average of a tail sequence. All our results are non-asymptotic and their consequences are discussed through a few applications.
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This paper aims to investigate the mathematical problem-solving capabilities of Chat Generative Pre-Trained Transformer (ChatGPT) in case of Bayesian reasoning. The study draws inspiration from Zhu & Gigerenzer's research in 2006, which posed the question: Can children reason the Bayesian way? In the pursuit of answering this question, a set of 10 Bayesian reasoning problems were presented. The results of their work revealed that children's ability to reason effectively using Bayesian principles is contingent upon a well-structured information representation. In this paper, we present the same set of 10 Bayesian reasoning problems to ChatGPT. Remarkably, the results demonstrate that ChatGPT provides the right solutions to all problems.
We propose and analyze a structure-preserving space-time variational discretization method for the Cahn-Hilliard-Navier-Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.
Benefiting from the development of deep learning, text-to-speech (TTS) techniques using clean speech have achieved significant performance improvements. The data collected from real scenes often contain noise and generally needs to be denoised by speech enhancement models. Noise-robust TTS models are often trained using the enhanced speech, which thus suffer from speech distortion and background noise that affect the quality of the synthesized speech. Meanwhile, it was shown that self-supervised pre-trained models exhibit excellent noise robustness on many speech tasks, implying that the learned representation has a better tolerance for noise perturbations. In this work, we therefore explore pre-trained models to improve the noise robustness of TTS models. Based on HIFI-GAN we first propose a representation-to-waveform vocoder, which aims to learn to map the representation of pre-trained models to the waveform. We then propose a text-to-representation Fastspeech2 model, which aims to learn to map text to pre-trained model representations. Experimental results on the LJSpeech and LibriTTS datasets show that our method outperforms those using speech enhancement methods in both subjective and objective metrics. Audio samples are available at: //zqs01.github.io/rep2wav/.
We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.
Deep needle insertion to a target often poses a huge challenge, requiring a combination of specialized skills, assistive technology, and extensive training. One of the frequently encountered medical scenarios demanding such expertise includes the needle insertion into a femoral vessel in the groin. After the access to the femoral vessel, various medical procedures, such as cardiac catheterization and extracorporeal membrane oxygenation (ECMO) can be performed. However, even with the aid of Ultrasound imaging, achieving successful insertion can necessitate multiple attempts due to the complexities of anatomy and tissue deformation. To address this challenge, this paper presents an innovative technology for needle tip real-time tracking, aiming for enhanced needle insertion guidance. Specifically, our approach revolves around the creation of scattering imaging using an optical fiber-equipped needle, and uses Convolutional Neural Network (CNN) based algorithms to enable real-time estimation of the needle tip's position and orientation during insertion procedures. The efficacy of the proposed technology was rigorously evaluated through three experiments. The first two experiments involved rubber and bacon phantoms to simulate groin anatomy. The positional errors averaging 2.3+1.5mm and 2.0+1.2mm, and the orientation errors averaging 0.2+0.11rad and 0.16+0.1rad. Furthermore, the system's capabilities were validated through experiments conducted on fresh porcine phantom mimicking more complex anatomical structures, yielding positional accuracy results of 3.2+3.1mm and orientational accuracy of 0.19+0.1rad. Given the average femoral arterial radius of 4 to 5mm, the proposed system is demonstrated with a great potential for precise needle guidance in femoral artery insertion procedures. In addition, the findings highlight the broader potential applications of the system in the medical field.
We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consist of derivative evaluations at the expansion point and the prior covariance kernel belongs to the class of Taylor kernels, which can be written in a certain power series form. We discuss and prove some results on maximum likelihood estimation of parameters of Taylor kernels. The proposed framework is a special case of Gaussian process regression based on data that is orthogonal in the reproducing kernel Hilbert space of the covariance kernel.
For the convolutional neural network (CNN) used for pattern classification, the training loss function is usually applied to the final output of the network, except for some regularization constraints on the network parameters. However, with the increasing of the number of network layers, the influence of the loss function on the network front layers gradually decreases, and the network parameters tend to fall into local optimization. At the same time, it is found that the trained network has significant information redundancy at all stages of features, which reduces the effectiveness of feature mapping at all stages and is not conducive to the change of the subsequent parameters of the network in the direction of optimality. Therefore, it is possible to obtain a more optimized solution of the network and further improve the classification accuracy of the network by designing a loss function for restraining the front stage features and eliminating the information redundancy of the front stage features .For CNN, this article proposes a multi-stage feature decorrelation loss (MFD Loss), which refines effective features and eliminates information redundancy by constraining the correlation of features at all stages. Considering that there are many layers in CNN, through experimental comparison and analysis, MFD Loss acts on multiple front layers of CNN, constrains the output features of each layer and each channel, and performs supervision training jointly with classification loss function during network training. Compared with the single Softmax Loss supervised learning, the experiments on several commonly used datasets on several typical CNNs prove that the classification performance of Softmax Loss+MFD Loss is significantly better. Meanwhile, the comparison experiments before and after the combination of MFD Loss and some other typical loss functions verify its good universality.
We propose a computationally and statistically efficient procedure for segmenting univariate data under piecewise linearity. The proposed moving sum (MOSUM) methodology detects multiple change points where the underlying signal undergoes discontinuous jumps and/or slope changes. Theoretically, it controls the family-wise error rate at a given significance level asymptotically and achieves consistency in multiple change point detection, as well as matching the minimax optimal rate of estimation when the signal is piecewise linear and continuous, all under weak assumptions permitting serial dependence and heavy-tailedness. Computationally, the complexity of the MOSUM procedure is $O(n)$ which, combined with its good performance on simulated datasets, making it highly attractive in comparison with the existing methods. We further demonstrate its good performance on a real data example on rolling element-bearing prognostics.
We tackle the problem of feature unlearning from a pre-trained image generative model: GANs and VAEs. Unlike a common unlearning task where an unlearning target is a subset of the training set, we aim to unlearn a specific feature, such as hairstyle from facial images, from the pre-trained generative models. As the target feature is only presented in a local region of an image, unlearning the entire image from the pre-trained model may result in losing other details in the remaining region of the image. To specify which features to unlearn, we collect randomly generated images that contain the target features. We then identify a latent representation corresponding to the target feature and then use the representation to fine-tune the pre-trained model. Through experiments on MNIST and CelebA datasets, we show that target features are successfully removed while keeping the fidelity of the original models. Further experiments with an adversarial attack show that the unlearned model is more robust under the presence of malicious parties.
Within the rapidly developing Internet of Things (IoT), numerous and diverse physical devices, Edge devices, Cloud infrastructure, and their quality of service requirements (QoS), need to be represented within a unified specification in order to enable rapid IoT application development, monitoring, and dynamic reconfiguration. But heterogeneities among different configuration knowledge representation models pose limitations for acquisition, discovery and curation of configuration knowledge for coordinated IoT applications. This paper proposes a unified data model to represent IoT resource configuration knowledge artifacts. It also proposes IoT-CANE (Context-Aware recommendatioN systEm) to facilitate incremental knowledge acquisition and declarative context driven knowledge recommendation.
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