mathematics of deep learning

More precisely, the work of [56] shows that the input to the network can be recovered from the network’s features at a certain layer if their size is proportional to the intrinsic dimension of the input data. Coding. In particular, researchers are exploiting geometry in general graphs via the spectrum of graph Laplacians and its spatial counterparts; see [55] for a recent survey on those advances. In a regression setting, the rows of Y∈RN×C denote the dependent variables for the rows of X. The classical approach to training neural networks is to minimize a (regularized) loss using backpropagation, , a gradient descent method specialized to neural networks. Batch normalization: Accelerating deep network training by reducing Architecture of a deep neural network(DNN) is shown in Fig 1. rep... 1097–1105. Surprisingly, these techniques from statistical physics are intimately connected to the regularization properties of partial differential equations (PDEs), problem, which penalizes greedy gradient descent. a comprehensive theory that explains the performance of deep networks as constructing optimal representations does not yet exist. Indeed, stationarity and stability to local translations are both leveraged in The gradient of the loss function at the minimum is zero and gradient descent algorithm tries to reach this point by iteratively making slow movements towards the direction of gradient. René Vidal acknowledges grant NSF 1618485. Assumptions (6)–(7) can be leveraged to approximate f from features Φ(X) that progressively reduce the spatial resolution. Traditional machine learning techniques require experts in a particular field to design feature extractors using domain knowledge. Sparse subspace clustering: Algorithm, theory, and applications. ∙ ChipHackers Authors are awesome people who joined hands with us to realize this goal. We are focused on sharing knowledge on electronics designing. Yoshua Bengio, Nicolas L Roux, Pascal Vincent, Olivier Delalleau, and Patrice Motifs assemble high level features like different parts of objects. Johns Hopkins University [2] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton, “Imagenet classification with deep convolutional neural networks,” in Advances in Neural Information Processing Systems 25, F. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger, Eds., pp. Pratik Chaudhari, Adam Oberman, Stanley Osher, Stefano Soatto, and Guillaume The moore-penrose pseudoinverse [7] is defined as (3). Thanks to the stationarity and local deformation priors, the linear operators at each layer have a constant number of parameters, independent of the input size. DNNs were able to reduce word error rates in speech recognition by 30% over conventional techniques [5]. Local entropy as a measure for sampling solutions in constraint The output (y) of the system is given by the multiplication of system matrix (A) and input vector (x). ∙ Your email address will not be published. In practice, it has been observed that if the size of the network is large enough, this strategy can lead to markedly different solutions for the network weights, which give nearly the same objective values and classification performance [22]. A block coordinate descent method for regularized multiconvex The loss surfaces of multilayer networks. neural networks. Notice that, while formally equivalent, the losses in (17) and (18) are conceptually different: In the former, the weights are a representation of the training set that is minimal and sufficient. For example, the state of a Kalman filter is an optimal representation for predicting data generated by a linear dynamical system with Gaussian noise; in other words, it is a minimal sufficient statistic for prediction. However sigmoid activation has low learning performance because the gradient decay to zero quickly. As a result of this extra structure, they are provably stable and locally invariant signal representations, and reveal the fundamental role of geometry and stability that underpins the generalization performance of modern deep convolutional network architectures; see Section IV. high-dimensional non-convex optimization. Specifically, [30, 31] analyzes the optimization problem in (2) using a purely deterministic approach which does not make any assumptions regarding the distribution of the input data, the network weight parameter statistics, or the network initialization. [3] Ronan Collobert and Jason Weston, “A unified architecture for natural language processing: Deep neural networks with multitask learning,” in Proceedings of the 25th International Conference on Machine Learning, New York, NY, USA, 2008, ICML ’08, pp. The information bottleneck loss can be re-written as the sum of a cross-entropy term, which is precisely the most commonly used loss in deep learning, with an additional regularization term. It is only recently that developments in theory have made this association possible, and developments in optimization and hardware have made deep neural networks a sensational success. The latter can be implemented by introducing noise, similar to adaptive dropout noise, in the learned representation [17]. is used as a balancing parameter. It is observed that a potential main goal of the training of the network is to treat the class boundary points while keeping the other distances approximately the same. 160–167, ACM. Therefore, the deeper the network, the stronger the shrinkage achieved. t... 2) Local deformations and scale separation: Tel Aviv University An important aspect for understanding better deep learning is the relationship between the structure of the data and the deep network. David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Rectifier nonlinearities improve neural network acoustic models. [1] Geoffrey Hinton, Li Deng, Dong Yu, George Dahl, Abdel-rahman Mohamed, Navdeep Jaitly, Andrew Senior, Vincent Vanhoucke, Patrick Nguyen, Brian Kingsbury, and Tara Sainath, “Deep neural networks for acoustic modeling in speech recognition,” IEEE Signal Processing Magazine, vol. Conf. Max pooling or mean pooling operation is performed to reduce the input data size. Communications on Pure and Applied Mathematics. International Conference on Machine Learning. Furthermore, many state-of-the-art networks are not trained with classical regularization, such as an ℓ1 or ℓ2 norm penalty on the weight parameters but instead rely on techniques such as dropout [11]. A. Choromanska, M. Henaff, M. Mathieu, G.B Arous, and Y. LeCun. AlexNet [2] was able to achieve top-5 error rate of 17.0% in ImageNet LSVRC-2010 contest outperforming other solutions by a significant margin. Tradeoffs between convergence speed and reconstruction accuracy in 2, are guaranteed not to exist. satisfaction problems. Next convolution layer learns mid level features like motifs created by combination of local edges. 02/12/2020 ∙ by Xavier Ouvrard, et al. The system matrix has the shape of. Each such operation, colloquially called a “layer”, consists of a linear transformation, say, a convolution of its input, followed by a point-wise nonlinear “activation function”, e.g., a sigmoid. Those three research directions originate from the viewpoint of statistical learning. Global optimality in tensor factorization, deep learning, and beyond. We conjecture that the two are related, and that the relation informs the generalization properties of the network, but specific bounds have yet been shown. Multilayer feedforward networks are universal approximators. For example, Table I shows the improvement achieved when using this regularization with the Wide ResNet architecture [74] for CIFAR-10 with different numbers of training examples. A key advantage of CNNs explaining their success in numerous tasks is that the geometric priors on which CNNs are based result in a sample complexity that avoids the curse of dimensionality. Identity mappings in deep residual networks. share, Recent work has studied the reasons for the remarkable performance of de... ensure that the operation is well-defined over, Structure of the data throughout a neural network, Towards an Information-Theoretic Framework. The problem of learning the network weights W is formulated as the following optimization problem: where ℓ(Y,Φ) is a loss function that measures the agreement between the true output, Y, and the predicted output, Φ(X,W) in (1), Θ is a regularization function designed to prevent overfitting, e.g., weight decay via ℓ2 regularization Θ(W)=∑Kk=1∥Wk∥2F, and λ>0 is a balancing parameter. In the vast majority of computer vision and speech analysis tasks, the unknown function f typically satisfies the following crucial assumptions: 1) Stationarity:

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