text stringlengths 1 7.76k | source stringlengths 17 81 |
|---|---|
188 8. PREPROCESSING AND UNSUPERVISED LEARNING Depending on the type of data at hand, a multitude of preprocessing steps can be considered. Some operations have little or no effect on the subsequent analysis. Others may seem natural but are far from trivial and can even be harmful. Here we will highlight only a few key ... | The+Shallow+and+the+Deep_Page_202_Chunk4101 |
8.1. NORMALIZATION AND TRANSFORMATIONS 189 8.1.1 Coordinate-wise transformations Frequently, coordinate- or feature-wise transformations are applied in order to harmonize the range or the statistical properties of the features. Centering and z-score transformation Given a set of observations or measurements resulting i... | The+Shallow+and+the+Deep_Page_203_Chunk4102 |
190 8. PREPROCESSING AND UNSUPERVISED LEARNING n ξ n log(ξ) Figure 8.1: Left panel: skewed histogram of a of a feature ξ that displays many small and relatively few large values in a given data set (illustration). Right panel: histogram of the log-transformed feature log(ξ). Min-Max feature scaling Sometimes it... | The+Shallow+and+the+Deep_Page_204_Chunk4103 |
8.1. NORMALIZATION AND TRANSFORMATIONS 191 Figure 8.2: Astrology as an artefact of implicit normalization by projecting stars onto the sur- face of a sphere. The right panel can also be viewed as an example of (mental) overfitting. Reproduced with kind permis- sion from John Atkinson for non-commercial use, see https://... | The+Shallow+and+the+Deep_Page_205_Chunk4104 |
192 8. PREPROCESSING AND UNSUPERVISED LEARNING is a superb example: unrelated stars appear to form meaningful clusters when they are implicitly projected onto a sphere, see Fig. 8.2 for a tongue-in-cheek illustration. As another example, L1-normalization using the so-called Manhattan norm yields transformed feature vec... | The+Shallow+and+the+Deep_Page_206_Chunk4105 |
8.2. DIMENSIONALITY REDUCTION 193 Figure 8.3: Left panel: schematic illustration of three-dimensional data points falling into a two-dimensional manifold. Right panel: the special case of a linear subspace or hyperplane that contains the data points. instance, a set of vectors {ξµ}P µ=1 could fall into (or close to) a ... | The+Shallow+and+the+Deep_Page_207_Chunk4106 |
194 8. PREPROCESSING AND UNSUPERVISED LEARNING ξ2 ξ3 ξ1 y1 y2 ||ξµ −ξν|| ||y µ −y ν|| Figure 8.4: Left panel: three-dimensional data points ξµ ∈R3 which display a relatively low variance in component ξ3 . Center: the two-dim. represen- tations y µ ∈R2 as obtained by metric MDS. Right panel: scatter plot of pair-w... | The+Shallow+and+the+Deep_Page_208_Chunk4107 |
8.2. DIMENSIONALITY REDUCTION 195 to optimize in MDS is the quadratic deviation E {y µ}P µ=1 = P µ,ν=1 µ<ν d µν N −dM(y µ, y ν) 2 . (8.7) All coordinates yµ j are considered degrees of freedom that can be obtained by minimization of E. A (local) minimum of (8.7) corresponds to an arrangement of P po... | The+Shallow+and+the+Deep_Page_209_Chunk4108 |
196 8. PREPROCESSING AND UNSUPERVISED LEARNING approximation and projection (UMAP) has become popular [MHM20]. Its key feature is the assumption that data is distributed in (or close to) a particular manifold along which distances are computed. Embedding methods are very popular, mainly in the context of data ex- plora... | The+Shallow+and+the+Deep_Page_210_Chunk4109 |
8.3. PCA AND RELATED METHODS 197 features in terms of the performance of trained regressors or classifiers. For each candidate set of features, training and potentially validation schemes have to be performed, which can result in considerable computational costs. An advantage of the wrapper approach over univariate filte... | The+Shallow+and+the+Deep_Page_211_Chunk4110 |
198 8. PREPROCESSING AND UNSUPERVISED LEARNING 8.3.1 Principal Component Analysis Due to their simplicity and intuitive nature, linear mappings are of particular interest and practical relevance. For example, Principal Component Analysis (PCA) is one of the most important and frequently used explicit mappings in the co... | The+Shallow+and+the+Deep_Page_212_Chunk4111 |
8.3. PCA AND RELATED METHODS 199 Restricting the search to normalized w with |w| = 1, it is straightforward to show that the maximum of Evar is achieved if w is the eigenvalue of C with the largest eigenvalue λ1: w = u1 with Cu1 = λ1u1. In case of degenerate leading eigenvalues, λ1 = λ2 = . . . = λm, we have to conside... | The+Shallow+and+the+Deep_Page_213_Chunk4112 |
200 8. PREPROCESSING AND UNSUPERVISED LEARNING Obviously the subspace of non-zero variability in the data set is at most P- dimensional in this case, as reflected by the rank of C. Hence, the maximum number of meaningful principal components is M = P. Typically, fewer com- ponents are used: under the assumption that the... | The+Shallow+and+the+Deep_Page_214_Chunk4113 |
8.3. PCA AND RELATED METHODS 201 Hence, assuming that 0 < λ2 < λ1, the weights w(t) will be dominated by the leading eigenvector u1 for t →∞. In case of degeneracies, the argument can be extended to the m-dimensional subspace of leading eigenvalues and the role of λ2 is taken over by λm+1 accordingly. We can obtain a m... | The+Shallow+and+the+Deep_Page_215_Chunk4114 |
202 8. PREPROCESSING AND UNSUPERVISED LEARNING Here terms O(η2) have been omitted. The iteration (8.22) is known as Oja’s Rule after Erkki Oja [Oja82]. It combines Hebbian learning with an appropriate weight decay that realizes an approximate normalization of w for small learning rates. Further principal components In ... | The+Shallow+and+the+Deep_Page_216_Chunk4115 |
8.3. PCA AND RELATED METHODS 203 The first terms in the brackets [. . .] of Eqs. (8.24, 8.25) correspond to the familiar Hebbian learning for linear units. The remaining terms can be motivated as approximate normalizations for k = m as in (8.20), while the terms with k ∕= m are associated with the pair-wise orthogonaliz... | The+Shallow+and+the+Deep_Page_217_Chunk4116 |
204 8. PREPROCESSING AND UNSUPERVISED LEARNING Figure 8.5: Left panel: three unimodal histograms with kurtosis ≈−1 (left panel), kurtosis ≈0 (center panel, corresponding to a normal density), and kurtosis ≈2 (right panel), respectively. As one example strategy for (b), we discuss here the maximization (in ab- solute va... | The+Shallow+and+the+Deep_Page_218_Chunk4117 |
8.4. CLUSTERING AND VECTOR QUANTIZATION 205 After a very brief discussion of elementary distance-based clustering meth- ods, we present two prominent and related methods of unsupervised data anal- ysis: Vector Quantization (VQ) by competitive learning and Gaussian Mixture Models (GMM) for density estimation. 8.4.1 Basi... | The+Shallow+and+the+Deep_Page_219_Chunk4118 |
206 8. PREPROCESSING AND UNSUPERVISED LEARNING 8.4.2 Competitive learning for Vector Quantization One possible aim of unsupervised learning is the representation of a potentially large set of feature vectors by a few typical representatives. The term Vector Quantization (VQ) has been coined for this task. In VQ, so-cal... | The+Shallow+and+the+Deep_Page_220_Chunk4119 |
8.4. CLUSTERING AND VECTOR QUANTIZATION 207 of Sec. 6.1, only the winner is adapted. However, in unsupervised VQ the winning prototype is always moved closer to the considered input vector; the update does not depend on additional information such as the labels in Learning Vector Quantization: Competitive learning (Vec... | The+Shallow+and+the+Deep_Page_221_Chunk4120 |
208 8. PREPROCESSING AND UNSUPERVISED LEARNING 8.4.3 Practical issues and extensions of VQ The quantization error HV Q can be interpreted as a quality criterion when comparing different prototype configurations. However, this is only meaningful for systems with the same number of prototypes. In general, HV Q will decreas... | The+Shallow+and+the+Deep_Page_222_Chunk4121 |
8.4. CLUSTERING AND VECTOR QUANTIZATION 209 (a) (b) (c) (d) Figure 8.6: Vector Quantization: representation of two-dimensional data points by prototypes (schematic). In each panel, 200 data points are displayed as small (red) dots, prototype positions corresponding to minimal HV Q are marked by filled black circles. The... | The+Shallow+and+the+Deep_Page_223_Chunk4122 |
210 8. PREPROCESSING AND UNSUPERVISED LEARNING Figure 8.7: Illustration of the elbow method: the quantization error (per sample) as obtained from the Iris flower data set with P = 150 feature vectors [Fis36] as a function of K in the K- means algorithm. K HV Q/P methods of clustering and the related criteria. In absence... | The+Shallow+and+the+Deep_Page_224_Chunk4123 |
8.5. DENSITY ESTIMATION 211 8.5 Density estimation A rather fundamental approach to obtaining insight into the properties of a given data set P = {ξµ}P µ=1 is to identify a model density that could have generated the observed data with high likelihood. Textbooks like [Bis95a,Bis06, HTF01] provide comprehensive overview... | The+Shallow+and+the+Deep_Page_225_Chunk4124 |
212 8. PREPROCESSING AND UNSUPERVISED LEARNING in terms of linear regression. Here, we only follow the maximum likelihood ap- proach and consider its application to a specific type of model densities in the next section. 8.5.2 Gaussian Mixture Models As in any machine learning problem, model selection is a key difficulty ... | The+Shallow+and+the+Deep_Page_226_Chunk4125 |
8.5. DENSITY ESTIMATION 213 It is straightforward to work out the necessary conditions for a maximum of the log-likelihood [Bis95a]. They can be written in the suggestive form wm = P µ=1 Qµ m Qµ m P ν=1 Qνm ξµ (8.37) σ2 m = P µ=1 Qµ m P ν=1 Qνm (ξµ −wm)2 (8.38) pm = 1 P P µ=1 Qµ m (8.39) with t... | The+Shallow+and+the+Deep_Page_227_Chunk4126 |
214 8. PREPROCESSING AND UNSUPERVISED LEARNING Figure 8.8: Illustration of density estimation by adaptation of a Gaussian Mixture Model. Left panel: 1000 two-dimensional data points drawn from a multimodal density. Center panel: initial density represented by a mixture of six Gaussians. Right panel: the model density a... | The+Shallow+and+the+Deep_Page_228_Chunk4127 |
8.6. MISSING VALUES AND IMPUTATION TECHNIQUES 215 In the considered limit, the GMM algorithm becomes identical with the intuitive K-means procedure presented in (8.31). This is yet another example for the observation made in Sec. 2.2 that many heuristic learning procedures can be interpreted as special cases or limits ... | The+Shallow+and+the+Deep_Page_229_Chunk4128 |
216 8. PREPROCESSING AND UNSUPERVISED LEARNING missingness from the provided data alone. The MCAR and MAR types, which are sometimes referred to as ignorable, are certainly the least difficult to handle. More systematic forms of missingness as in MNAR, would require sophisticated modelling techniques to take them properl... | The+Shallow+and+the+Deep_Page_230_Chunk4129 |
8.6. MISSING VALUES AND IMPUTATION TECHNIQUES 217 ξj by the corresponding mean or median in the data set as computed from all instances in which ξj is present. This can, however introduce or enhance a bias in imbalanced data sets. As a seemingly more sophisticated choice, the class-wise mean or median of ξj could be us... | The+Shallow+and+the+Deep_Page_231_Chunk4130 |
218 8. PREPROCESSING AND UNSUPERVISED LEARNING imputed by regression and the entire procedure is repeated until all missing fea- tures (and naive estimates) have been replaced by regression based imputation. The imputation of all missing values can be done with different selections of the initial feature ξk and by varyi... | The+Shallow+and+the+Deep_Page_232_Chunk4131 |
8.7. OVER- AND UNDERSAMPLING, AUGMENTATION 219 be achieved by randomly selecting the same number of examples from each class. Hence, the training set size will be limited to at most min{P1, P2, . . . PC} examples per class. Undersampling can be limited to the actual training set while validation and test sets can remai... | The+Shallow+and+the+Deep_Page_233_Chunk4132 |
220 8. PREPROCESSING AND UNSUPERVISED LEARNING SMOTE (Synthetic Minority Oversampling Technique) - randomly select an example from the minority class - determine its k nearest neighbors in the same class, e.g. according to Euclidean distance - select one of the neighbors with equal probability 1/k - generate a new data... | The+Shallow+and+the+Deep_Page_234_Chunk4133 |
8.7. OVER- AND UNDERSAMPLING, AUGMENTATION 221 is discussed briefly in [GBC16] with additional references given. One survey on image data augmentation for Deep Learning is provided in [SK19]. Obviously, some of the risks mentioned in Sec. 8.7.3 are also relevant for data augmentation. In principle, it is not possible to... | The+Shallow+and+the+Deep_Page_235_Chunk4134 |
222 8. PREPROCESSING AND UNSUPERVISED LEARNING | The+Shallow+and+the+Deep_Page_236_Chunk4135 |
Concluding quote Everybody right now, they look at the current technology, and they think, “OK, that’s what artificial neural nets are.” And they don’t realize how arbitrary it is. We just made it up! And there’s no reason why we shouldn’t make up some- thing else. — Geoffrey Hinton | The+Shallow+and+the+Deep_Page_237_Chunk4136 |
224 | The+Shallow+and+the+Deep_Page_238_Chunk4137 |
Appendix A Optimization There is nothing objective about objective functions. — James L. McClelland Here we summarize some essential mathematical concepts concerning real- valued functions of multi-dimensional arguments and their optimization. In particular, we consider local extrema and gradient-based search strategie... | The+Shallow+and+the+Deep_Page_239_Chunk4138 |
226 A. OPTIMIZATION Figure A.1: Real world illustration of extrema and saddle points of a func- tion in d = 2 dimensions (elevation z(x, y)). Zero gradients can correspond to maxima, minima, saddle points or extended flat regions. Photo taken on the Mt. Whitney trail (https://en.wikipedia.org/wiki/Mount_Whitney_Trail). ... | The+Shallow+and+the+Deep_Page_240_Chunk4139 |
A.2. LOCAL EXTREMA AND SADDLE POINTS 227 A.2 Local extrema and saddle points First, we consider unconstrained problems of the form minimize x ∈Rd f(x) with the real-valued objective function f. Obviously, analogous results are ob- tained for local maxima by considering the minima of −f(x). First, we assume that x can b... | The+Shallow+and+the+Deep_Page_241_Chunk4140 |
228 A. OPTIMIZATION Note also that an eigenvalue ρi relates to the curvature (second derivative) of the function f in x∗along the direction of the normalized eigenvector ui. Defining fi(γ) = f(x∗+ γui) and using u⊤ i ui = 1 we have fi(γ) ≈(x∗) + (γui)⊤H∗(γui) = fi(0) + γ2 ρ∗ i , i.e. ∂2 fi ∂γ2 γ=0 = ρ... | The+Shallow+and+the+Deep_Page_242_Chunk4141 |
A.2. LOCAL EXTREMA AND SADDLE POINTS 229 where the last term is independent of w. We proceed as in (2.2.2) by considering the first order, necessary conditions (A.6) for a solution w∗: ∇wESSE = [χ⊤χ]w∗−χ⊤Y != 0. (A.15) In Sec. 2.2.2 we have already presented the formal solution of (A.15) in terms of the left pseudoinver... | The+Shallow+and+the+Deep_Page_243_Chunk4142 |
230 A. OPTIMIZATION A.3 Constrained optimization A.3.1 Equality constraints Frequently, one encounters optimization problems of the form minimize x ∈Rd f(x) subject to n equality constraints {gi(x) = 0}n i=1 , (A.18) where the real-valued functions gi define additional conditions under which f(x) has to be minimized: Th... | The+Shallow+and+the+Deep_Page_244_Chunk4143 |
A.3. CONSTRAINED OPTIMIZATION 231 A.3.2 Example: under-determined linear equations We revisit the set of linear equations considered in Sec. A.2.2 χ w = Y. (A.21) If it represents P < N equations for the N unknowns w ∈RN, the system can have many solutions. Then, a number io of vectors vi ∕= 0 of χ exist with χvi = 0. ... | The+Shallow+and+the+Deep_Page_245_Chunk4144 |
232 A. OPTIMIZATION Remark: A unified treatment Unsolvable over-determined and solvable under-determined systems can be treated in a unified way [BH12]. Consider the limits lim γ→0+ [χ⊤χ + γ IN]−1 χ⊤ and lim γ→0+ χ⊤[χ χ⊤+ γ IP ]−1 (A.25) with the P-dim. and N-dim. identity matrices IP and IN, respectively. Both limits ex... | The+Shallow+and+the+Deep_Page_246_Chunk4145 |
A.3. CONSTRAINED OPTIMIZATION 233 where we use the shorthand (...)|∗for the evaluation in x = x∗and {λi = λ∗ i } . The first condition (A.27) corresponds to the stationarity of the Lagrangian with respect to the variables x. Condition (A.28) simply represents the original constraints, while (A.29) states that all multip... | The+Shallow+and+the+Deep_Page_247_Chunk4146 |
234 A. OPTIMIZATION As outlined in Sec. 3.7, the first condition shows that the Lagrange multipliers play the role of embedding strengths. The weights can be interpreted as to result from iterative Hebbian learning and can, in fact, be eliminated from the optimization problem. A.3.4 The Wolfe Dual for convex problems Th... | The+Shallow+and+the+Deep_Page_248_Chunk4147 |
A.4. GRADIENT BASED OPTIMIZATION 235 Example: minimize x∈Rn 1 2x⊤C x subject to Cx ≥b. (A.36) A particular property of this example problem is that the same matrix C de- fines the quadratic form and the linear constraints. More general examples for the application of duality are presented in, e.g., [Fle00, PAH... | The+Shallow+and+the+Deep_Page_249_Chunk4148 |
236 A. OPTIMIZATION Taylor expansion (A.2), the corresponding directional derivative in xo can be written as lim α→0 f(xo + αa) −f(x) α = a⊤∇f(xo). Obviously the conventional partial derivatives are recovered by setting a = ei, i.e. by taking the directional derivative along the coordinate unit vectors ei with ei k = δ... | The+Shallow+and+the+Deep_Page_250_Chunk4149 |
A.4. GRADIENT BASED OPTIMIZATION 237 Introducing the shorthand δt = x(t) −x∗for the deviation from the optimum, the update (A.39) at step t of the descent satisfies approximately δt ≈δt−1 −ηH∗δt−1 ≈[I −ηH∗] δt−1 ≈[I −ηH∗]2 δt−2 . . . ≈[I −ηH∗]t δ0 with the N-dim. identity matrix I. Here, we have subtracted x∗on both sid... | The+Shallow+and+the+Deep_Page_251_Chunk4150 |
238 A. OPTIMIZATION The fact that convergence can be achieved with a constant, non-zero value of η constitutes an important insight. However, the practical usefulness of this insight is limited: the properties of the Hessian can be very different for every local minimum and they are obviously not available in advance. T... | The+Shallow+and+the+Deep_Page_252_Chunk4151 |
A.5. VARIANTS OF GRADIENT DESCENT 239 For a more detailed discussion and an introduction of the so-called co-variant or natural gradient we refer to the lecture notes of M. Toussaint as a starting point [Tou12]. The corresponding method of Natural Gradient Descent can be related to information theoretic metrics and was... | The+Shallow+and+the+Deep_Page_253_Chunk4152 |
240 A. OPTIMIZATION Essentially, the structure of the algorithm is the same as for a conventional gradient descent step when it is performed as a loop over the d coordinates. Here, however, in each component we make use of the previously updated coordinates, while in conventional gradient descent the values from the pr... | The+Shallow+and+the+Deep_Page_254_Chunk4153 |
A.5. VARIANTS OF GRADIENT DESCENT 241 A corresponding cost function can be written in the form E(W) = 1 P P µ=1 eµ(W) where eµ(W) = e(ξµ, yµ| W) (A.46) quantifies the contribution of an individual example data to the total costs. Quite generally, the vector W is meant to concatenate all degrees of freedom in the trai... | The+Shallow+and+the+Deep_Page_255_Chunk4154 |
242 A. OPTIMIZATION On average over the random selection process, an SGD update is guided by the true negative gradient −∇W E and, therefore, we can expect that the cost functions typically decreases over many steps for suitable choices of η. However, single training steps may actually increase the objective function... | The+Shallow+and+the+Deep_Page_256_Chunk4155 |
A.6. EXAMPLE CALCULATION OF A GRADIENT 243 Sophisticated schemes have been devised in which the learning rate is not ex- plicitly time dependent, but is adapted in the course of training. The adaptation can be based on (estimated) second order derivatives or on the observed variance of the gradient over several update ... | The+Shallow+and+the+Deep_Page_257_Chunk4156 |
244 A. OPTIMIZATION The term vm g′(. . .) corresponds to the derivative of the only term in the sum K j=1 . . . that contains the weight vector w(m). Finally the factor ξn appears because w(m) · ξ = N j=1 w(m) j ξj and thus ∂(w(m) · ξ) ∂w(m) n = ξn. Note that the r.h.s. of Eqs. (A.52,A.53) are just numbers as they... | The+Shallow+and+the+Deep_Page_258_Chunk4157 |
List of figures 1.1 Neurons and synapses . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Action potentials and firing rate . . . . . . . . . . . . . . . . . . 5 1.3 Sigmoidal activation functions . . . . . . . . . . . . . . . . . . . . 7 1.4 Recurrent neural networks . . . . . . . . . . . . . . . . . . . . . . 11 1... | The+Shallow+and+the+Deep_Page_259_Chunk4158 |
. . . . . . . . . . . . 97 4.5 SVM: Illustration of the non-linear transformation . . . . . . . . 99 5.1 Generic layered network . . . . . . . . . . . . . . . . . . . . . . . 108 245 | The+Shallow+and+the+Deep_Page_259_Chunk4159 |
246 LIST OF FIGURES 5.2 Interval selection by sigmoidal functions . . . . . . . . . . . . . . 110 5.3 Selection of ROI in high dimensions . . . . . . . . . . . . . . . . 111 5.4 Constructed network for universal function approximation . . . . 112 5.5 Soft Committee Machine . . . . . . . . . . . . . . . . . . . . . . . ... | The+Shallow+and+the+Deep_Page_260_Chunk4160 |
. . . . . . . . . . . . . . . . . . . . . 194 8.5 Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.6 Vector Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.7 Elbow method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.8 Gaussian Mixtures . . . . . . . . ... | The+Shallow+and+the+Deep_Page_260_Chunk4161 |
List of algorithms Adaline (parallel updates), 70 Adaline (sequential updates), 71 AdaTron (sequential updates), 76 AdaTron with errors (sequential updates), 88 Batch Gradient Descent (basic form), 117 Competitive learning (Vector Quantization), 207 Cross validation (n-fold), 173 Gaussian Mixture Model, maximum likelih... | The+Shallow+and+the+Deep_Page_261_Chunk4162 |
Abbrev. and acronyms Adaline adaptive linear element, adaptive linear neuron AdaTron adaptive perceptron (algorithm) AUC area under the curve AUROC area under the receiver operating characteristics curve BAC balanced accuracy CM committee machine CNN convolutional neural network CoD coefficient of determination ELM extre... | The+Shallow+and+the+Deep_Page_262_Chunk4163 |
PCA principal component analysis PCT perceptron convergence theorem PR precision-recall Prec precision PSP perceptron storage problem RBF radial basis functions Rec recall ReLU rectified linear unit r.h.s. right hand side ROC receiver operating characteristics ROI region of interest RSLVQ robust soft learning vector qua... | The+Shallow+and+the+Deep_Page_263_Chunk4164 |
Notes on the bibliography References are sorted alphabetically by their BIBTEX keys for ease of brows- ing. The BIBTEX source file is available upon request from the author or at www.cs.rug.nl/˜biehl. Most online sources point to the publisher’s final versions, some of which might not be publicly available. Where possibl... | The+Shallow+and+the+Deep_Page_264_Chunk4165 |
Bibliography [AB89] J.K. Anlauf and M. Biehl. The AdaTron: an adaptive perceptron algo- rithm. Europhys. Lett., 10(7):687–692, 1989. Online: https://iopscience. iop.org/article/10.1209/0295-5075/10/7/014, see also Europhys. Lett. 11(4):387 for an Erratum: https://doi.org/10. 1209/0295-5075/11/4/016. [ABR64] M. A. Aizer... | The+Shallow+and+the+Deep_Page_265_Chunk4166 |
252 BIBLIOGRAPHY of Proceedings of Machine Learning Research, pages 3–13, 2009. Online: https://proceedings.mlr.press/v8/airola10a.html. [ASFL11] M.J. Azur, E.A. Stuart, C. Frangakis, and P.J. Leaf. Multiple imputa- tion by chained equations: what is it and how does it work? Interna- tional Journal of Methods in Psychi... | The+Shallow+and+the+Deep_Page_266_Chunk4167 |
BIBLIOGRAPHY 253 [BCLC15] M. Boareto, J. Cesar, V. Leite, and N. Caticha. Supervised variational relevance learning, an analytic geometric feature selection with applica- tions to omic data sets. IEEE/ACM Trans. Computational Biology and Bioinformatics, 12(3):705–711, 2015. Online: https://www.researchgate. net/publica... | The+Shallow+and+the+Deep_Page_267_Chunk4168 |
254 BIBLIOGRAPHY 7(2):92–111, 2016. Online: https://pure.rug.nl/ws/portalfiles/portal/ 172538141/wcs.1378.pdf. [Bie17] M. Biehl. Biomedical applications of prototype based classifiers and rel- evance learning. In D. Figueiredo, C. Martin-Vide, D. Pratas, and M.A. Vega-Rodriguez, editors, AlCoB: 4th International Conferen... | The+Shallow+and+the+Deep_Page_268_Chunk4169 |
BIBLIOGRAPHY 255 [BMR+20] T.B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhari- wal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, S. Agarwal, A. Herbert-Voss, G. Krueger, T. Henighan, R. Child, A. Ramesh, D.M. Ziegler, J. Wu, C. Winter, C. Hesse, M. Chen, E. Sigler, M. Litwin, S. Gray, B. Chess, J. Clark, ... | The+Shallow+and+the+Deep_Page_269_Chunk4170 |
256 BIBLIOGRAPHY [Bur89] P. Burman. A comparative study of ordinary cross-validation, v- fold cross validation and the repeated learning testing-model methods. Biometrika, 76:503–514, 1989. Online: https://www.jstor.org/stable/ 2336116. [BW88] E.B. Baum and F. Wilczek. Supervised learning of probability distri- butions... | The+Shallow+and+the+Deep_Page_270_Chunk4171 |
BIBLIOGRAPHY 257 [CMB00] J.L. Castro, C.J. Mantas, and J.M. Benitez. Neural networks with a con- tinuous squashing function in the output are universal approximators. Neural Networks Letter, 13:561–563, 2000. Online: https://sci2s.ugr.es/ sites/default/files/ficherosPublicaciones/0820_2000-benitez-NN.pdf. [Cov65] T.M. Co... | The+Shallow+and+the+Deep_Page_271_Chunk4172 |
258 BIBLIOGRAPHY [Die00] R. Dietrich. Statistical Mechanics of Neural Networks: Enhancement by Weighting of Examples. PhD thesis, Julius-Maximilians-Universität Würzburg, Germany, 2000. Online: https://citeseerx.ist.psu.edu/doc/ 10.1.1.15.4844. [DLR77] A.P. Dempster, N.M. Lair, and D.B. Rubin. Maximum Likelihood from I... | The+Shallow+and+the+Deep_Page_272_Chunk4173 |
BIBLIOGRAPHY 259 [Faw06] T. Fawcett. An introduction to ROC analysis. Pattern Recogni- tion Letters, 27:861–874, 2006. Online: https://www.researchgate.net/ publication/222511520_Introduction_to_ROC_analysis. [FCC98] T. Friess, N. Cristianini, and C. Campbell. The Kernel-AdaTron Al- gorithm: a Fast and Simple Learning ... | The+Shallow+and+the+Deep_Page_273_Chunk4174 |
260 BIBLIOGRAPHY [Gal90] S.I. Gallant. Perceptron-based learning algorithms. IEEE Transac- tions on Neural Networks, 1(2):179–191, 1990. Online: https://www. ling.upenn.edu/courses/Fall_2007/cogs501/Gallant1990.pdf. [GBB11] X. Glorot, A. Bordes, and Y. Bengio. Deep sparse rectifier neural networks. In G. Gordon, D. Duns... | The+Shallow+and+the+Deep_Page_274_Chunk4175 |
BIBLIOGRAPHY 261 [GT18] A.N. Gorban and I.Y. Tyukin. Blessing of dimensionality: mathematical foundations of the statistical physics of data. Phil. Trans. R. Soc. A, 376(20170237), 2018. Online: https://doi.org/10.1098/rsta.2017.0237. [Gue97] K. Guerney. An Introduction to Neural Networks. UCL Press, Lon- don, 1997. On... | The+Shallow+and+the+Deep_Page_275_Chunk4176 |
262 BIBLIOGRAPHY In M. Gertz and B. Ludäscher, editors, Scientific and Statisti- cal Database Management, pages 482–500, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. Preprint: https://imada.sdu.dk/~zimek/ publications/SSDBM2010/SNN-SSDBM2010-preprint.pdf. [HKP91] J.A. Hertz, A.S. Krogh, and R.G. Palmer. Introdu... | The+Shallow+and+the+Deep_Page_276_Chunk4177 |
BIBLIOGRAPHY 263 [HT01] D.J. Hand and R.J. Till. A simple generalisation of the area under the ROC curve for multiple class classification problems. Machine Learning, 45(2):171–186, 2001. Online: https://link.springer.com/article/10.1023/ A:1010920819831. [HTF01] T. Hastie, R. Tibshirani, and J. Friedman. The Elements o... | The+Shallow+and+the+Deep_Page_277_Chunk4178 |
264 BIBLIOGRAPHY [KHV14] M. Kaden, W. Hermann, and T. Villmann. Optimization of general statistical accuracy measures for classification based on learning vector quantization. In M. Verleysen, editor, Proc. of the European Symposium on Artificial Neural Networks ESANN 2014, pages 47–52. i6doc.com, 2014. Online: https://c... | The+Shallow+and+the+Deep_Page_278_Chunk4179 |
BIBLIOGRAPHY 265 [LBD+89] Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hub- bard, and L. D. Jackel. Backpropagation Applied to Handwritten Zip Code Recognition. Neural Computation, 1(4):541–551, 12 1989. Online: https://doi.org/10.1162/neco.1989.1.4.541. [LBH18] Y. LeCun, Y. Bengio, and G. Hinton. D... | The+Shallow+and+the+Deep_Page_279_Chunk4180 |
266 BIBLIOGRAPHY [LR02] R.J.A. Little and D.B. Rubin. Statistical Analysis with Missing Data. Wiley Series in Probability and Statistics. John Wiley & Sons, 2 edition, 2002. Online: https://doi.org/10.1002/9781119013563. [LV07] J.A. Lee and M. Verleysen. Nonlinear Dimensionality Reduction. Springer, 2007. Online: https... | The+Shallow+and+the+Deep_Page_280_Chunk4181 |
BIBLIOGRAPHY 267 [MM92] H.N. Mhaskar and C.A. Micchelli. Approximation by Superposition of Sigmoidal and Radial Basis Functions. Adv. in Appl. Mathemat- ics, 13:350–373, 1992. Online: https://www.sciencedirect.com/science/ article/pii/019688589290016P. [MN89] M. Mezard and J.-P. Nadal. Learning in feedforward layered n... | The+Shallow+and+the+Deep_Page_281_Chunk4182 |
268 BIBLIOGRAPHY [ND91] D. Nabutovsky and E. Domany. Learning the unlearnable. Neural Com- putation, 3(4):604–616, 1991. Online: https://doi.org/10.1162/neco. 1991.3.4.604. [NE14] D. Nova and P.A. Estévez. A review of Learning Vector Quantization classifiers. Neural Computing and Applications, 25(3-4):511–524, 2014. Pre... | The+Shallow+and+the+Deep_Page_282_Chunk4183 |
BIBLIOGRAPHY 269 [Oja82] E. Oja. Simplified neuron model as a principal component analyzer. J. Math. Biology, 15:267–273, 1982. Online: https://doi.org/10.1007/ BF00275687. [Oja89] E. Oja. Neural Networks, Principal Components, and Subspaces. International Journal of Neural Systems, 1(1):61–68, 1989. Online: https://www... | The+Shallow+and+the+Deep_Page_283_Chunk4184 |
270 BIBLIOGRAPHY [PJ92] B.T. Polyak and A.B. Juditsky. Acceleration of stochastic approxima- tion by averaging. SIAM J. Control Optim., 30(4):838–855, 1992. On- line: https://www.researchgate.net/publication/236736831. [PJS17] J. Peters, D. Janzing, and B. Schölkopf. Elements of Causal Inference: Foundations and Learni... | The+Shallow+and+the+Deep_Page_284_Chunk4185 |
BIBLIOGRAPHY 271 [RHW86] D.E. Rumelhart, G.E. Hinton, and R.J. Williams. Learning represen- tations by back-propagating errors. Nature, 323:533–536, 1986. Online: https://www.nature.com/articles/323533a0. [RKSV20] J. Ravichandran, M. Kaden, S. Saralajew, and T. Villmann. Variants of dropconnect in learning vector quant... | The+Shallow+and+the+Deep_Page_285_Chunk4186 |
272 BIBLIOGRAPHY ’16, pages 1135–1144, New York, NY, USA, 2016. Association for Com- puting Machinery. Online: https://www.kdd.org/kdd2016/papers/files/ rfp0573-ribeiroA.pdf. [Ruj93] P. Ruján. A fast method for calculating the perceptron with maximal stability. J. de Physique I, 3(2):277–290, 1993. Online: https://hal. ... | The+Shallow+and+the+Deep_Page_286_Chunk4187 |
BIBLIOGRAPHY 273 [Sch15] J. Schmidhuber. Deep learning in neural networks: an overview. Neural Networks, 61:85–117, 2015. Preprint: https://arxiv.org/abs/1404.7828. [Sej20] T.J. Sejnowski. The unreasonable effectiveness of deep learning in ar- tificial intelligence. Proc. National Academy of Sciences of the Unit- ede Sta... | The+Shallow+and+the+Deep_Page_287_Chunk4188 |
274 BIBLIOGRAPHY [SO03] S. Seo and K. Obermayer. Soft Learning Vector Quantization. Neural Computation, 15:1589–1604, 2003. Online: https://www.researchgate. net/publication/10698895_Soft_Learning_Vector_Quantization. [SR98] D. Saad and M. Rattray. Learning with regularizers in multilayer neural networks. Physical Revi... | The+Shallow+and+the+Deep_Page_288_Chunk4189 |
BIBLIOGRAPHY 275 [SY98] A.S. Sato and K. Yamada. An analysis of convergence in Generalized LVQ. In L. Niklasson, M. Bodén, and T. Ziemke, editors, Interna- tional Conference on Artificial Neural Networks, ICANN’98, pages 172– 176. Springer, Berlin, 1998. Online: https://link.springer.com/chapter/ 10.1007/978-1-4471-1599... | The+Shallow+and+the+Deep_Page_289_Chunk4190 |
276 BIBLIOGRAPHY Applications (ICMLA), pages 7–10. IEEE, New York, NY, 2012. Online: https://ieeexplore.ieee.org/document/6406802. [VL63] V.N. Vapnik and A.Y. Lerner. Recognition of patterns with help of generalized portraits. Avtomat. i Telemekh., 24(6):774–780, 1963. On- line: https://www.mathnet.ru/php/archive.phtml... | The+Shallow+and+the+Deep_Page_290_Chunk4191 |
BIBLIOGRAPHY 277 [Wid60] B. Widrow. An Adaptive "Adaline" Neuron Using Chemical "Memis- tors", 1960. Technical Report No. 1553-2. Online: https://www-isl. stanford.edu/~widrow/papers/t1960anadaptive.pdf. [Wid12] B. Widrow. Youtube channel widrowlms, 2012. Online: https://www. youtube.com/user/widrowlms. [Wik22] Wikiped... | The+Shallow+and+the+Deep_Page_291_Chunk4192 |
278 BIBLIOGRAPHY D. McAllester, editors, Proceedings of the 30th International Confer- ence on Machine Learning, volume 28 of Proceedings of Machine Learn- ing Research, pages 1058–1066, Atlanta, Georgia, USA, 2013. PMLR. Online: https://proceedings.mlr.press/v28/wan13.pdf. [YCC18] M. Yamada, J. Chen, and Y. Chang. Tra... | The+Shallow+and+the+Deep_Page_292_Chunk4193 |
The Shallow and the Deep is a collection of lecture notes that offers an accessible introduction to neural networks and machine learning in general. However, it was clear from the beginning that these notes would not be able to cover this rapidly changing and growing field in its entirety. The focus lies on classical m... | The+Shallow+and+the+Deep_Page_294_Chunk4194 |
University of Arkansas, Fayetteville University of Arkansas, Fayetteville ScholarWorks@UARK ScholarWorks@UARK Open Educational Resources 2-8-2019 University Physics I: Classical Mechanics University Physics I: Classical Mechanics Julio Gea-Banacloche University of Arkansas, Fayetteville Follow this and additional works... | University Physics I Classical Mechanics_Page_1_Chunk4195 |
University Physics I: Classical Mechanics Julio Gea-Banacloche | University Physics I Classical Mechanics_Page_2_Chunk4196 |
Cover image from NASA, https://www.nasa.gov/image-feature/jpl/not-really-starless-at-saturn | University Physics I Classical Mechanics_Page_3_Chunk4197 |
University Physics I: Classical Mechanics Julio Gea-Banacloche First revision, Fall 2019 This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. Developed thanks to a grant from the University of Arkansas Libraries | University Physics I Classical Mechanics_Page_4_Chunk4198 |
ii | University Physics I Classical Mechanics_Page_5_Chunk4199 |
Contents Preface i 1 Reference frames, displacement, and velocity 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Particles in classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Aside: the atomic perspective . . . . . . . . . . . ... | University Physics I Classical Mechanics_Page_6_Chunk4200 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.