The Stability of Inverse Problems

By Mario Bertero, Christine De Mol, G. Viano. 1980

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Citations in the corpus

Natural continual learning: success is a journey, not (just) a destination (2021, methodology): “While this has not been explored in the context of projection-based continual learning, it will be interesting to combine NCL with methods such as Tikhonov dampening (Tikhonov, 1943) in future work to automatically adjust α and make NCL a hyperparameter-free method.”
New Insights and Perspectives on the Natural Gradient Method (2014, methodology): “Examples include Tikhonov regularization/damping and the closely related trustregion method (e.g. Tikhonov, 1943; Moré and Sorensen, 1983; Conn et al., 2000; Nocedal and Wright, 2006), and other ones such as the “structural damping” approach of Martens and Sutskever (2011), or the approach present…”
Learning Deep Parsimonious Representations (2016, background): “Standard neural network regularization involves penalties on the weights based on the norm of the parameters [29, 30].”
Improving Neural Networks with Dropout (2013, background): “These include L2 weight decay (more generally Tikhonov regularization [24]), lasso [23] and KL-sparsity regularization which minimizes the KL-divergence between the distribution of hidden unit activations and”
Probabilistic numerics and uncertainty in computations (2015, unknown): “2In fact, the very same connection between least-squares estimation and Gaussian inference has been re-discovered repeatedly, simply because least-squares estimation has been re-discovered repeatedly after Legendre, under names such as ridge regression [5], Kriging [6], Tikhonov’s method [7] and so ”
The stability for the Cauchy problem for elliptic equations (2009, background): “As is well-known since Tikhonov [95] the modern notion of stability for ill-posed problems (also called conditional stability) consists of estimating the dependence upon the data of the unknown solution of the problem at hand, when an a-priori bound on the solution itself is available.”
Artificial Neural Networks for Neuroscientists: A Primer (2020, background): “Adding a L2 regularization term, L reg = λ ij W 2 ij , to the loss function [Tikhonov, 1943] (equivalent to weight decay [Krogh and Hertz, 1992]) discourages the network from using large connection weights.”
Estimation Stability With Cross-Validation (ESCV) (2013, methodology): “For example, in ﬁtting the usual linear regression model, the Lasso (Tibshirani 1996) and ridge regression (Tikhonov 1943; Hoerl 1962) adds a L 1 and L 2 penalty on the coeﬃcient estimates respectively to the usual least squares ﬁt objective function.”
Regularized estimation of image statistics by Score Matching (2010, methodology): “The regularizer is related to Tikhonov regularization [22] and curvature-driven smoothing [2] where the square of the curvature of the energy surface at the data points are also penalized.”
Deep Learning Techniques for Inverse Problems in Imaging (2020, methodology): “Examples of this framework include Tikhonov regularization [54], sparsity regularization in some basis or frame [55], [56], and total variation regularization [11], [57].”
Learnability, Stability and Uniform Convergence (2010, methodology): “In particular, adding ‖h‖(2) is the so-called Tikhonov Regularization technique, which has been known for more than half a century (see Tikhonov (1943)).”
Long Short-Term Memory Recurrent Neural Network for Remaining Useful Life Prediction of Lithium-Ion Batteries (2018, methodology): “L1 and L2 regularization methods [36], [37] are the two most widely used methods to prevent neural networks from overfit-”
Using Lasso for Predictor Selection and to Assuage Overfitting: A Method Long Overlooked in Behavioral Sciences (2015, methodology): “Although the focus of this paper is on Lasso, ridge regression (Hoerl & Kennard, 1970; Tikhonov, 1943) is a natural predecessor of Lasso that provides a foundation for discussing Lasso and also maintains some advantages in its own right that are worth mentioning.”
Machine Learning Advances for Time Series Forecasting (2020, methodology): “The ridge regression is also known as a particular case of Tikhonov Regularization (Tikhonov, 1943, 1963; Tikhonov & Arsenin, 1977), in which the scale matrix is diagonal with identical entries.”
: "Statistical Modeling: The Two Cultures" (Statistical Science, 2001, 16(3), 199-231) (unknown): “Paradoxically, Tikhonov regularization (Tikhonov, 1943) characterizes much of the early nonparametrics works (Hoerl and Kennard, 1970), and it also characterizes current theoretical underpinnings of deep neural networks; e.g., Jacot et al. (2018); Arora et al. (2019).”
Analyzing Neuroimaging Data Through Recurrent Deep Learning Models (2018, methodology): “To reduce the risk of overfitting, resulting from the large number of model coefficients, the whole-brain lasso applies Least Absolute Shrinkage regularization to the likelihood function of the logistic model (Tikhonov, 1943; Tibshirani, 1996).”