A Semi-Supervised Predictive Sparse Decomposition Based on Task-Driven Dictionary Learning

Stochastic differential equation (SDE) models are the foundation for pricing and hedging financial derivatives. The drift and volatility functions in SDE models are typically chosen to be algebraic functions with a small number ( < 5 ) of parameters which can be calibrated to market data. A more flexible approach is to use neural networks to model the drift and volatility functions, which provides more degrees-of-freedom to match observed market data. Training of models requires optimizing over an SDE, which is computationally challenging. For European options, we develop a fast stochastic gradient descent (SGD) algorithm for training the neural network-SDE model. Our SGD algorithm uses two independent SDE paths to obtain an unbiased estimate of the direction of steepest descent. For American options, we optimize over the corresponding Kolmogorov partial differential equation (PDE). The neural network appears as coefficient functions in the PDE. Models are trained on large datasets (many contracts), requiring either large simulations (many Monte Carlo samples for the stock price paths) or large numbers of PDEs (a PDE must be solved for each contract). Numerical results are presented for real market data including S&P 500 index options, S&P 100 index options, and single-stock American options. The neural-network-based SDE models are compared against the Black-Scholes model, the Dupire's local volatility model, and the Heston model. Models are evaluated in terms of how accurate they are at pricing out-of-sample financial derivatives, which is a core task in derivative pricing at financial institutions. Specifically, we calibrate a neural network-SDE model to market data for a financial derivative on an asset with price S t with a payoff function g ( s ) , and we then evaluate its generalization accuracy for a financial derivative on the same asset S t but with a different payoff function f ( s ) . In addition to comparing out-of-sample pricing accuracy, we evaluate the hedging performance of the neural network-SDE model.

The sparse coding method is formulated as an information theoretic optimization problem. The rate distortion theory leads to an objective functional which can be interpreted as an information theoretic formulation of the sparse coding. Viewing as an entropy minimization problem, the rate distortion theory and consequently the sparse coding are extended to discriminative variants. As a concrete example of this information theoretic sparse coding, a discriminative non-linear sparse coding algorithm with neural networks is proposed. Experimental results of gender classification by face images show that the discriminative sparse coding is more robust to noise, compared to the conventional method which directly uses images as inputs to a linear support vector machine.

We consider in this work a system of two stochastic differential equations named the perturbed compositional gradient flow. By introducing a separation of fast and slow scales of the two equations, we show that the limit of the slow motion is given by an averaged ordinary differential equation. We then demonstrate that the deviation of the slow motion from the averaged equation, after proper rescaling, converges to a stochastic process with Gaussian inputs. This indicates that the slow motion can be approximated in the weak sense by a standard perturbed gradient flow or the continuous-time stochastic gradient descent algorithm that solves the optimization problem for a composition of two functions. As an application, the perturbed compositional gradient flow corresponds to the diffusion limit of the Stochastic Composite Gradient Descent (SCGD) algorithm for minimizing a composition of two expected-value functions in the optimization literatures. For the strongly convex case, such an analysis implies that the SCGD algorithm has the same convergence time asymptotic as the classical stochastic gradient descent algorithm. Thus it validates, at the level of continuous approximation, the effectiveness of using the SCGD algorithm in the strongly convex case.

Linear prediction methods, such as least squares for regression, logistic regression and support vector machines for classification, have been extensively used in statistics and machine learning. In this paper, we study stochastic gradient descent (SGD) algorithms on regularized forms of linear prediction methods. This class of methods, related to online algorithms such as perceptron, are both efficient and very simple to implement. We obtain numerical rate of convergence for such algorithms, and discuss its implications. Experiments on text data will be provided to demonstrate numerical and statistical consequences of our theoretical findings.

In the paper, we consider a stochastic gradient descent algorithm in combination with a dropout method. We used the theory of intuitionistic fuzzy sets for the assessment of the equivalence of the respective assessment units. We also consider a degree of uncertainty when the information is not enough.

This paper considers blind deconvolution problem that to recover unknown signals fand g from their convolution signal. Non-convex optimization approach is an efficient method to get the solution, but it is a challenge to find the exact solution for a non-convex optimization problem. Existing work provides a full gradient descent (GD) method converging to the global minimum from a proper initialization. However, GD algorithm is not computationally efficient. In this paper, we design the first stochastic gradient descent (SGD) algorithm that converges linearly to the exact solution. We also design a Kaczmarz algorithm which adapts the step size of SGD algorithm. It also has the linear convergence and is more computationally efficient. Finally, we analyze the global geometry of the objective function. Although the function is non-convex, its expectation has a good geometry that every local minimum is also a global optimal point. Our numerical experiments demonstrate that both SGD and Kaczmarz algorithms are more computationally efficient and can converge to the global minimum even without a proper initialization.

Efficiency of large-scale learning is a hot topic in both academic and industry. The stochastic gradient descent (SGD) algorithm, and its extension mini-batch SGD, allow the model to be updated without scanning the whole data set. However, the use of approximate gradient leads to the uncertainty issue, slowing down the decreasing of objective function. Furthermore, such uncertainty may result in a high frequency of meaningless update on the model, causing a communication issue in parallel learning environment. In this work, we develop a batch-adaptive stochastic gradient descent (BA-SGD) algorithm, which can dynamically choose a proper batch size as learning proceeds. Particularly on the basis of Taylor extension and central limit theorem, it models the decrease of objective value as a Gaussian random walk game with rebound. In this game, a heuristic strategy of determining batch size is adopted to maximize the utility of each incremental sampling. By evaluation on multiple real data sets, we demonstrate that by smartly choosing the batch size, the BA-SGD not only conserves the fast convergence of SGD algorithm but also avoids too frequent model updates.

This work investigates the problem of efficiently learning discriminative low-dimensional (LD) representations of multiclass image objects. We propose a generic end-to-end approach that jointly optimizes sparse dictionary and convolutions for learning LOW-dimensional discriminative image representations, named SparConvLow, taking advantage of convolutional neural networks (CNNs), dictionary learning, and orthogonal projections. The whole learning process can be summarized as follows. First, a CNN module is employed to extract high-dimensional (HD) preliminary convolutional features. Second, to avoid the high computational cost of direct sparse coding on HD CNN features, we learn sparse representation (SR) over a task-driven dictionary in the space with the feature being orthogonally projected. We then exploit the discriminative projection on SR. The whole learning process is consistently treated as an end-to-end joint optimization problem of trace quotient maximization. The cost function is well-defined on the product of the CNN parameters space, the Stiefel manifold, the Oblique manifold, and the Grassmann manifold. By using the explicit gradient delivery, the cost function is optimized via a geometrical stochastic gradient descent (SGD) algorithm along with the chain rule and the backpropagation. The experimental results show that the proposed method can achieve a highly competitive performance with the state-of-the-art (SOTA) image classification, object categorization, and face recognition methods, under both supervised and semi-supervised settings. The code is available at https://github.com/MVPR-Group/SparConvLow.

Structural health monitoring (SHM) provides an economic approach which aims to enhance understanding the behavior of structures by continuously collecting data through multiple networked sensors attached to the structure. These data are then utilized to gain insight into the health of a structure and make timely and economic decisions about its maintenance. The generated SHM sensing data are non-stationary and exist in a correlated multi-way form which makes the batch/off-line learning and standard two-way matrix analysis unable to capture all of these correlations and relationships. In this sense, the online tensor data analysis has become an essential tool for capturing underlying structures in higher-order datasets stored in a tensor $${\mathcal {X}} \in {\mathbb {R}} ^{I_1 \times \cdots \times I_N} $$ X ∈ R I 1 × ⋯ × I N . The CANDECOMP/PARAFAC (CP) decomposition has been extensively studied and applied to approximate X by N loading matrices $$A(1),\ldots ,A(N)$$ A ( 1 ) , … , A ( N ) where N represents the order of the tensor. We propose a novel algorithm, FP-CPD, to parallelize the CANDECOMP/PARAFAC (CP) decomposition of a tensor $${\mathcal {X}} \in {\mathbb {R}} ^{I_1 \times \cdots \times I_N} $$ X ∈ R I 1 × ⋯ × I N . Our approach is based on stochastic gradient descent (SGD) algorithm which allows us to parallelize the learning process, and it is very useful in online setting since it updates $${\mathcal {X}}^{t+1}$$ X t + 1 in one single step. Our SGD algorithm is augmented with Nesterov’s accelerated gradient and perturbation methods to accelerate and guarantee convergence. The experimental results using laboratory-based and real-life structural datasets indicate fast convergence and good scalability.

Recent research has demonstrated that median filtering can be adopted as a postprocessing procedure to cover the tampering traces in many image forgeries. Hence, median filtering detection has become an important branch in digital image forensics. Most existing median filtering forensic techniques detect median filtering traces using handcraft features, but their performance degrades with smaller image size and stronger compression. To get more detailed features, automatic feature learning becomes a viable approach. In this paper, a discriminative multi-scale sparse coding based approach is proposed. This approach performs sparse coding representation in multi-scale patch domain to automatically extract distinct patterns caused by median filtering at different levels. To further improve the representative ability of sparse codes, discriminative dictionaries for various patch sizes are derived from labeled training patches of specific sizes. Based on the discriminative sparse codes, max-pooling is then employed over all corresponding patches to obtain representative features for each specific patch size. The final features are derived by concatenating all the features of various patch sizes. The experiment results show that the proposed approach out-performs three state-of-the-art approaches for median filter detection in most cases.

We provide parallel algorithms for large-scale matrix completion on problems with millions of rows, millions of columns, and billions of revealed entries. We focus on in-memory algorithms that run either in a shared-memory environment on a powerful compute node or in a shared-nothing environment on a small cluster of commodity nodes; even very large problems can be handled effectively in these settings. Our ASGD, DSGD-MR, DSGD++, and CSGD algorithms are novel variants of the popular stochastic gradient descent (SGD) algorithm, with the latter three algorithms based on a new "stratified SGD" approach. All of the algorithms are cache-friendly and exploit thread-level parallelism, in-memory processing, and asynchronous communication. We investigate the performance of both new and existing algorithms via a theoretical complexity analysis and a set of large-scale experiments. The results show that CSGD is more scalable, and up to 60 % faster, than the best-performing alternative method in the shared-memory setting. DSGD++ is superior in terms of overall runtime, memory consumption, and scalability in the shared-nothing setting. For example, DSGD++ can solve a difficult matrix completion problem on a high-variance matrix with 10M rows, 1M columns, and 10B revealed entries in around 40 min on 16 compute nodes. In general, algorithms based on SGD appear to perform better than algorithms based on alternating minimizations, such as the PALS and DALS alternating least-squares algorithms.

In this paper, we present a handwritten character recognition (HCR) system that aims to recognize first-order logic handwritten formulas and create editable text files of the recognized formulas. Dense feedforward neural networks (NNs) are utilized, and their performance is examined under various training conditions and methods. More specifically, after three training algorithms (backpropagation, resilient propagation and stochastic gradient descent) had been tested, we created and trained an NN with the stochastic gradient descent algorithm, optimized by the Adam update rule, which was proved to be the best, using a trainset of 16,750 handwritten image samples of 28 × 28 each and a testset of 7947 samples. The final accuracy achieved is 90.13%. The general methodology followed consists of two stages: the image processing and the NN design and training. Finally, an application has been created that implements the methodology and automatically recognizes handwritten logic formulas. An interesting feature of the application is that it allows for creating new, user-oriented training sets and parameter settings, and thus new NN models.

Biased outcomes in machine learning models can arise due to various factors, including limited training data, imbalanced class distribution, suboptimal training methodologies, and overfitting. In training neural networks with Stochastic Gradient Descent (SGD) and backpropagation, the choice of hyperparameters like learning rate and momentum is crucial to influencing the model’s performance. A comprehensive grid search study was conducted using static hyperparameters with standard SGD and dynamic hyperparameters with the Adam optimizer. The investigation focused on a multifaceted analysis across different tasks —- classification, segmentation, and detection — and was applied to four image-based applications: digit classification using the MNIST dataset, tuberculosis detection from chest X-ray images, lung segmentation in chest X-ray images, and the detection of malaria parasites in blood smear images. In the first comparative study on the MNIST dataset, the SGD algorithm consistently outperformed the Adam algorithm as noise levels increased. SGD held a slight advantage in accuracy over Adam in a noise-free environment. This advantage became more apparent as noise was introduced and increased to moderate levels. At high noise levels, both algorithms experienced a significant decline in performance, yet SGD maintained a relatively better accuracy compared to Adam. This trend underscored SGD’s superior ability to generalize across varying noise conditions. For TB detection in the second application, the DenseNet121 architecture was used, and it was found that Adam showed better performance on the larger TBX11K dataset. However, SGD outperformed Adam on a smaller subset of TBX11K. In the third comparative study for lung segmentation using COVID-19 and NLM datasets, SGD slightly outperformed Adam based on the mean and Hausdorff distances. In the case of malaria parasite detection using the YOLOv8 architecture, SGD and Adam optimizers showed varying performances across different conditions. Initially, both achieved high accuracies on the full dataset, with Adam slightly outperforming SGD. However, with decreasing dataset size, SGD maintained more consistent performance, while Adam’s accuracy fluctuated significantly. In noise tests, both showed equal accuracy on clean data, but under noise conditions, SGD maintained higher accuracies, suggesting better generalization capabilities. Further experiments assessed how well SGD and Adam could cope with domain shifts, mainly when using data from different countries. SGD’s generalization performance was superior to Adam’s performance in these experiments. In summary, although Adam is arguably the more popular of the two optimization techniques, SGD held its own in the experiments and showed superior performance when it came to generalization or classifying in noisy conditions.

Coherent beam combining (CBC) of fiber laser array is a promising way to achieve high output power. Phase control is one key point to implement CBC. Appropriate feedback structures should be established to achieve phase control. Most feedback structures of CBC are established after the lasers emit to free space and consist of a set of lenses or mirrors. Those optical elements in free space may hinder array size and integration. In this paper, we demonstrated an all-fiber structured CBC method with distributed phase-locking. By adding an all-fiber measurement loop beside the main laser chain, the phase of main laser chain is appended to the measuring loop. Phases of each main laser chain are locked indirectly though the measurement loops by using stochastic parallel gradient descent (SPGD) algorithm. The principle of distributed phase-locking is also illustrated. Corresponding simulations are carried out and two-channel fiber lasers are coherently combined by this method. The experimental results show that the structure can achieve phase-locking effectively. Stable and distinct interference fringe is observed. Additionally, the structure proposed in this paper is straightforwardly building and expanding.

This study investigates the iterative convergences of neural network for prediction turning tool wear. For the smart manufacturing, the intelligent prediction systems have been gradually developing for processing of CNC machine tools. Recently, many artificial intelligent algorithms of machine learning have been widely applied for forecasting and decision making in intelligent manufacturing. In general, the cutting tool wear in manufacturing of CNC machine tool plays a major role for a high quality and an efficient operation, but it is very difficult to diagnose and prognoses the tool wear for tool life due to many cutting parameters. Therefore, the study investigates the iterative gradient convergences of backpropagation neural network (BNN) algorithm for prediction tool life with analytics of its convergence and stability. The estimative methods of iterative convergences include stochastic gradient descent (SGD), momentum, adaptive gradient (Adagrad), adaptive delta (Adadelta), and adaptive moment (ADAM) algorithms. In BNN prediction model, the data inputs are the cutting speed, feed rate, and total material removal volume and data output is tool wear measured from the microscope. Results showed that the tool wear curves at different cutting conditions can be predicted and trained using BNN model for intelligent manufacturing. In addition, the convergence of ADAM gradient for the tool wear in all cases is the best prediction for the BNN model. However, it is worth to notice that the momentum gradient is faster training speed to converge to a constant error at fewer iteration numbers.