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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 techniques and selected popular methods: ◦Frequently, some form of normalization or coordinate transformation is performed which then facilitates the application of a particular machine learning framework. This can account for mismatched scaling or skewed distributions of feature values, for instance. ◦Dimensionality reduction plays a key role in learning problems, where the (nominal) dimension of feature vectors is relatively high compared to their intrinsic dimension and/or to the number of example data. The iden- tification of specifically two- or three-dimensional representations plays an important role in the exploration and in the human-understandable vi- sualization of data sets or the trained systems. Feature selection can be interpreted as a specific form of dimensionality reduction, aiming at a the identification of a subset of available features that is suitable and sufficient for the given task. ◦Unsupervised density estimation, Vector Quantization and cluster- ing can provide important insights into the structure of the data. For example, the detection of pronounced clusters of data in a preprocessing step can help to design a specific classifier or regression system. ◦Often, the imputation of missing values in incomplete data sets is nec- essary, in particular when data is scarce and incomplete feature vectors cannot be simply discarded. ◦Under- and oversampling techniques are applied when data sets are imbalanced with respect to the classes in a supervised learning problem. Similarly, data augmentation aims at enriching the training data in order to achieve better robustness against noise or to impose certain in- variances in the training process, e.g. by generating rotated or shifted training images in object recognition. The above goals and methods are obviously interrelated. For example, dimen- sionality reduction by Principal Component Analysis obviously constitutes also a coordinate transformation. Similarly, density estimation can be used for the imputation of missing values. 8.1 Normalization and transformations Probably the most frequently used preprocessing steps concern some form of normalization or transformation of the feature vectors prior to the actual ma- chine learning analysis. Clearly, such operations should never be applied blindly without evaluating their effect on the subsequent analysis. | 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 in N-dim. features vectors IP = ξ µ ∈RNP µ=1 , it can be convenient and/or useful to subtract the mean observed in the data set. As a result, one obtains centered feature vectors ξµ = ξ µ −m where m = 1 P P µ=1 ξ µ with 1 P P µ=1 ξµ = 0. (8.1) Here, we subtract the empirical mean m as observed in the data set. In (rare) cases where the true mean of the corresponding probability density is known, it could be used to replace the empirical one. In the N-dimensional Euclidean space, the centering (8.1) corresponds to a simple translation of all data points, which has no effect on, for instance, pair-wise Euclidean distances or angles defined by triplets of data points. A slightly more involved transformation is frequently applied in order to obtain zero mean, unit variance features: zµ j = ξ µ j −mj σj with σ2 j = 1 P P µ=1 ξ µ j −mj 2 (8.2) Here, mj is the empirical mean of component ξ µ j in IP as defined in Eq. (8.1) and σj is the corresponding standard deviation. As a result, the so-called standard scores or z-scores zµ j display zero mean and unit variance in the given data set. Hence, zµ j quantifies by how many standard deviations a feature value differs from the empirical mean in the data set. Positive (negative) zµ j correspond to above (below) average values, respectively. The z-score transformation or standardization is used to account for features that scale differently, which can be detrimental in a supervised or unsupervised analysis of the data. The transformation renders the representation independent of linear rescaling of individual features and choice of units of measure (as in miles or centimeters for the length of an object). On a univariate level, considering only single features, the effect of the monotonic transformation is not critical. However, one should be aware that it can alter the relations between features in a non-trivial way. As one example, a z-score transformation can affect the outcome of a Vector Quantization scheme, see the discussion of Fig. 8.6 in Sec. 8.4.3. A variety of similar linear transformations can be motivated for specific problems and data sets. For instance, we could employ the feature median and interquartile range (IQR) for a scaling analogous to (8.2) in order to achieve zero median and unit IQR data. | 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 is desirable to transform features to a fixed range of values, e.g. reflecting the specific requirements of the machine learning system in use. With minj = min ξµ j P µ=1 and maxj = max ξµ j P µ=1 we can achieve that ξj = ξj −minj maxj −minj ∈[0, 1]. (8.3) Note that the use of minimum and maximum values can be very sensitive to the presence or absence of extreme values in a data set. This or similar transformations are occasionally also referred to as normal- ization in the literature. In order to avoid confusion, we will use the term only in the context of the actual Lp-normalization of vectors as described in the following subsection. Non-linear feature transformations The effect of linear feature-wise transformations can be highly non-trivial, for instance in the context of classification or unsupervised clustering, cf. Sec. 8.4. Obviously, the impact of non-linear transformations can be even greater. A large variety of such transformations can be considered, having specific goals in mind and taking into account domain knowledge about the properties of the data set at hand. As just one particular example we discuss briefly the popular log-transform. Often, features of the considered data display a skewed distribution of values. The illustration in Figure 8.1 (left panel) shows the histogram of a non-negative feature which frequently assumes small values and only rarely larger ones. In such cases, a feature-wise logarithmic transformation of the form ξj = log(ξj) or ξj = log(ξj + ) with > 0 (8.4) | 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:// wronghands1.com/ for more of his brilliant cartoons. reshapes the histogram to appear – loosely speaking – more Gaussian.2 The transformed data is exactly Gaussian only if the original data follows a log-normal distribution of the form P(ξj) ∝exp −(log ξj −µ)2 2σ2 . (8.5) While this is rarely exactly the case in real world data, skewed distributions similar to the histogram in Fig. 8.1 (left) are not uncommon in practice, e.g. in the context of medical data relating to bio-markers, see [Bie17] for an example. In any case, which transformations make sense in a particular problem de- pends on available domain knowledge and insights into the data structure. 8.1.2 Normalization One of the most frequently applied type of transformation is the normalization of observed N-dimensional feature vectors. Based on so-called Lp-norms ||ξ||p = N j=1 | ξj |p 1/p we can consider vectors ξ = ξ ||ξ||p, (8.6) which then all display Lp-norm one. For p = 2, corresponding to the familiar Euclidean norm, we obtain vectors of the same length in feature space. In other words, all data points are projected onto the unit sphere in RN. Using Euclidean distance after L2- normalization is equivalent to measuring distances in terms of angles between the original feature vectors. The L2-normalization appears to be a natural thing to do, for instance in the context of the Perceptron, see Chapter 3. In more general contexts, however, the effects of normalization can be non- trivial and even lead to artefacts and mis-interpretations of the data. Astrology 2The choice = 1 is often used for non-negative features as it maps ξj = 0 to ξj = 0. | 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 vectors with j |ξj| = 1. For non-negative features e.g. representing amounts of chemical components in a sample, the normalized data can be interpreted as concentrations. Similarly, event counts would be transformed to normalized frequencies or probabilities. Normalization as a preprocessing step can be helpful and greatly beneficial in practical problems. In any case, it should be applied based on available domain knowledge, and its effects on the subsequent analysis should be carefully evaluated. 8.2 Dimensionality reduction The low-dimensional representation of high-dimensional data plays an impor- tant role in the context of data analysis and machine learning, see e.g. [Bis06, HTF01, LV07, MPH09, BBH12] for a variety of methods and concepts. Several interconnected, overlapping motivations for dimension reduction can be identi- fied: ◦Preprocessing If high-dimensional data are presented to a machine learning system, the number of adaptive quantities will be (at least) of the same order. This may hinder successful training, in particular if the number of available examples is limited. Dimension reduction can help to overcome this diffi- culty. ◦Exploration Low-dimensional representations help to explore a given data set prior to, say, supervised learning. Linear or non-linear projections can reveal structures in the data, e.g. clusters or subspaces in which the feature vectors are located. Such insights can help to design appropriate systems in the subsequent analysis by taking specific properties of the data into account. ◦Visualization Closely related to the previous point, two- or three-dimensional represen- tations provide visualizations of the data set which facilitate interaction with the user or domain expert. Before further analysis, visualization can give useful insight into the structure of the problem. In retrospect, e.g. after the training of a classifier, visualization helps to evaluate its perfor- mance and provides information about regions where classes overlap or individual misclassified data points are located [SHH20]. It is important to realize that the intrinsic dimensionality of feature vectors ξ ∈RN does not necessarily coincide with the nominal dimension N. For | 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 low-dimensional manifold M ⊂RN as illustrated in Fig. 8.3. A particular simple example is a linear subspace, i.e. a hyperplane which contains the linear dependent feature vectors, an illustration is shown in the right panel of Fig. 8.3. Note that the popular term curse of dimensionality [Bel57] is avoided here, because (a) it is not obvious that a nominally high dimension is necessarily detri- mental for the performance of machine learning methods (see e.g. [HKK+10, GT18]) and (b) superstition brings bad luck. We can distinguish two essentially different concepts for the low-dimensional representation of high-dim. data: in one family of approaches, each original data point is represented by an individual counter-part in a low-dim. latent space without requiring a parameterized mapping between the spaces. The positions of the representatives are directly determined by means of optimizing a suitable cost function which is based on the aim of (approximately) preserving neigh- borhood relations, pair-wise distances, or the overall topology of the original data points. These approaches do not require an explicit linear or non-linear functional mapping from the high- to the low-dimensional space. As an impor- tant example, we present Multi-dimensional Scaling (MDS) below and briefly mention a few other popular methods. In the second major framework, an explicit mapping is determined which provides linear or non-linear projections of the original data into the latent space. The projection is optimized according to a specific criterion which is evaluated w.r.t. a given data set. While these methods are less flexible due to the pre-defined form of the actual mapping, they offer the possibility to project out-of-sample data after training. The most popular example of the basic concept is the well-known Principal Component Analysis (PCA) which is discussed in Sec. 8.3 also from a neural network perspective. | 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-wise Euclidean distances in N =3 dimensions vs. distances in M =2 after MDS. 8.2.1 Low-dimensional embedding Consider a given set of N-dim. feature vectors IP = ξµ ∈RNP µ=1 with a distance measure dN(ξ, ξ′) that quantifies the pair-wise dissimilarity of data points in IP. As a straightforward example we can think of the Euclidean distance in N dimensions, but the consideration of any meaningful dissimilarity is possible. We refer to the pair-wise distances as d µν N = dN(ξµ, ξν) in the following. A number of methods aim at finding an embedding of the data points in a low-dimensional Euclidean vector space that preserves relations between the individual feature vectors in the original space, as much as possible. The goal could be to approximately reproduce the pair-wise distances themselves, their rank structure, or associated probability densities. 8.2.2 Multi-dimensional Scaling The goal of metric Multi-dimensional Scaling (MDS) 3 is to find representatives y µ ∈RM with M < N, which preserve pair-wise distances and their relations as far as possible. To this end we consider coordinates yµ j ∈R (j = 1, . . . M and µ = 1, . . . P) in the target space. There, we evaluate the dissimilarities of the corresponding vectors {y µ}P µ=1 by means of a metric dM(y, y ′). Again, this measure could be based on any reasonable vector norm, with Euclidean metric being the classical and by far most popular choice. Note that, in general, the measures dN and dM need not be of the same type. An example cost function 3So-called classical MDS is also known as Principal Coordinates Analysis and is not dis- cussed here [Mea92]. | 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 points in M dimensions which reflects the pair-wise distances d µν N as well as possible. For M < N it is in general not possible to obtain a perfect solution with E = 0. Obviously, we can also not expect to find a unique minimum of E, the actual outcome of MDS will depend on the initial configuration of y µ and on the poten- tial randomness in the training process. Figure 8.4 illustrates MDS in terms of a simple example based on Euclidean distance in both spaces: three-dimensional feature vectors ξµ (left panel) display a relatively small variance in component ξ3. The corresponding two-dimensional representations y µ, obtained by min- imizing the quadratic deviation (8.7), is displayed in the center panel. The scatter plot in the right panel shows that very similar pair-wise distances have been achieved in M = 2 dimensions. Quantitative measures, e.g. a correlation coefficient, could be obtained in order to evaluate the quality of the MDS result. Note that the quality of the embedding is invariant under, e.g., simultaneous translations or rotations of the y µ. The term MDS is frequently meant to imply the use of Euclidean distances in RN and RM, and minimizing the quadratic deviation (8.7). Various modifica- tions of the basic idea have been suggested and applied in practice. Obviously, a variety of distance measures dN and dM could be considered. Moreover, spe- cific cost functions can be chosen which put, for instance, different emphasis on small or large distances. A good overview of MDS related methods can be obtained from [LV07] and [MPH09]. In particular [BBH12] discusses several choices in a unified framework, including the so-called Isomap [TdSL00], Sam- mon mapping [Sam69], Local Linear Embedding (LLE) [RS00], and Laplacian Eigenmaps [BN03]. 8.2.3 Neighborhood Embedding Another popular family of methods is often used for the visual inspection of complex data sets. In the original Stochastic Neighborhood Embedding (SNE) as introduced by Hinton and Roweis [HR03], the data is characterized in terms of Gaussian probabilities in the original and in the embedding space. These are replaced by long-tailed student-t distributions in the popular t-distributed SNE (t-SNE) that was later suggested by van der Maaten and Hinton [MH08]. Both, SNE and t-SNE, aim at the minimization of the Kullback-Leibler divergence as a measure of (dis-)similarity between the assumed densities in the original and the embedding space. The optimization can be done using gradient-based methods. Recently, an alternative known as Uniform manifold | 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- ploration and visualization. For more detailed presentations of neighborhood embedding, we refer the reader to the original literature and reviews like [LV07, BBH12,BBK20]. One of the main drawbacks of these methods is that, generally speaking, the embedding coordinates y are not directly interpretable and their connection to the original feature space is often unclear. In this sense, embed- ding methods are not very appealing as preprocessing steps for further super- vised learning (regression or classification). Moreover, out-of-sample extension, i.e. the post-hoc embedding of data points that were not in IP is non-trivial. One option is to add them to the set and then recompute all y µ on the basis of the extended data, which is obviously very costly. Alternatively, one can try to obtain an explicit mapping function Ψ that approximately realizes Ψ : ξµ →y µ and can be applied to novel data points, see for instance [EHT20] for a Deep Learning based approach. 8.2.4 Feature selection A rather direct way of reducing the dimensionality of input data is to select a subset of features, neglecting all others. It appears to be a natural idea to simply try all different combinations of features and select the best possible subset. In practice, however, the number of subsets with k features selected from N dimensions is N k and grows very rapidly with k and N. An overview of basic feature selection methods can be found in [GE03,CS14, JBB15]. Three main strategies for the selection of feature subsets have been considered in the literature: ◦Filter methods: So-called filter methods aim at the identification of useful features with- out actually training a classification or regression model. In unsupervised approaches, features are selected or rejected based on their statistical prop- erties, independent of the actual target problem. For instance, correlations between different features could be exploited in order to discard redundant features in a given data set. Supervised filtering takes into account the label information in the data. For instance, in the context of regression, individual features can be evalu- ated and selected in terms of their correlations with the target. Similarly, in classification problems, mutual information or cross entropy between features and the target can serve as a selection criterion. Most frequently, these criteria are applied in a univariate fashion, feature by feature. This bears the risk of missing the relevance of combinations of features which would be revealed in an appropriate multivariate analysis. ◦Wrapper methods: In the wrapper approach, the idea is to consider different combinations of | 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 filter meth- ods is that, in principle, favorable combinations of features can be found. Obviously, there is no straightforward way to find the optimal set of fea- tures for a given task. One can start with the full set of available features and remove single ones from the set, in every step selecting the one that has the least impact on the performance after training. Alternatively, one could add features to the set, following a greedy strategy to improve the quality of the classification or regression in every step. ◦Embedded methods: In embedded methods the selection of a subset of features is an integral part of training a specific type of model, i.e. classifier or regression system. A popular example would be the training of decision trees in a Random Forest [Bre01], in which a tree selects a particular feature in every branch- ing step. Hence, the actual feature set is compiled while the system is trained. While this can be more efficient than the wrapper approach, it is - in a sense - limited to an essentially univariate evaluation of features. 8.3 PCA and related projection methods A variety of methods derives an explicit mapping of the form Ψ : RN →RM with y = Ψ(ξ) ∈RM (8.8) directly from the data set IP. The mapping Ψ is parameterized and obtained in a data driven process. Hence, unlike MDS or other embedding methods, the mapping can be applied to out-of-sample data ξ /∈P immediately. This is particularly important for methods of supervised learning, where we can derive Ψ from the training data and apply the same mapping to novel data in the working phase. In principle, we can obtain a meaningful projection by parameterizing the function Ψ in a suitable way and then optimizing its parameters according to an appropriate cost function. All criteria discussed in the context of embedding in the previous section could be employed for the identification of an explicit mapping as well. We discuss here only methods which employ linear projections of the data. Most methods, including Principal and Independent Component Analysis can be extended in various ways. Non-linear versions can be formulated, for in- stance, in terms of shallow or deep neural autoencoders as discussed in Sec. 5.5.1. Kernelized versions also exist, see for instance [SSM98,LG19] and refer- ences therein. | 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 context of data analysis, unsupervised learning, low-dimensional rep- resentation and visualization. Primarily, PCA is employed to determine low- dimensional representations. In addition it can be used to identify supposedly non-informative contributions in a given data set. The basic idea is to disregard linear combinations of features that hardly vary over the observed data. Like many other standard and well-established methods, PCA can be mo- tivated and derived from a variety of perspectives. Here we follow a mostly heuristic line of thoughts and point out the relation to the theoretical back- grounds in passing. For convenience we will assume throughout the following that the data set used for the identification of the mapping Ψ is centered, i.e. 1 P P µ=1 ξµ = 0. If this has been achieved by a centering of the form (8.1), the same transformation using the empirical mean in IP has to be performed on novel input vectors, before Ψ can be applied. The direction w = u1 ∈RN along which the centered data displays the largest variance is of particular interest. We can formulate the search as an optimization problem with respect to the empirical variance along w: Evar = 1 P P µ=1 (yµ)2 with the projections yµ = w · ξµ. (8.9) It is plausible to assume that the corresponding solution of largest variance, is the direction in which most of the information about ξ is contained. In fact, this is rigorously true under the assumption that all features follow a normal distribution. If we define the empirical covariance matrix 4 C ∈RN×N with elements Cij = 1 P P µ=1 ξµ i ξµ j , (8.10) we can rewrite the objective function, Eq. (8.9), as a quadratic form Evar = N j,k=1 wj wk 1 P P µ=1 ξµ j ξµ k = w⊤C w. (8.11) The matrix C is positive semi-definite by definition, the quadratic form is non- negative but otherwise unbounded. We can assume that C has ordered eigen- values λ1 ≥λ2 ≥. . . ≥λN ≥0. (8.12) 4not to be confused with the complementary P ×P matrix C with elements Cµν ∝ j ξµ j ξν j defined in (3.74) in the context of classification. | 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 consider the corresponding m-dimensional eigenspace. Note that the variance associated with the normalized u1 is given directly by its eigenvalue: 1 P P µ=1 (u1 · ξµ)2 = u⊤ 1 Cu1 = λ1(u1)2 = λ1. (8.13) Once we have determined u1 as the direction of largest variation, we proceed by identifying the direction u2 which displays the largest variance in the space orthogonal to u1. In general we can determine the ordered sequence of directions um in which the data displays the largest variance with um · uk for all k = 1, 2 . . . m −1. Hence, we identify the M leading Principal Components of IP with the leading eigenvectors of C {wk}M k=1 with projections yµ k = wk · ξµ and variance 1 P P µ=1 (yµ k)2 = λk. (8.14) In the simple illustration shown in Fig. 8.3 (right panel), the vectors marked in red correspond to the two leading eigenvectors or principal components. They can serve as coordinate axes defining the two-dimensional subspace of largest variation in the data set. The M-dim. vectors y µ = (yµ 1 , yµ 2 , . . . yµ M)⊤serve as M-dimensional repre- sentations of the data set. The associated linear subspace displays the largest variances in the data set. Under appropriate assumptions of Gaussianity, this implies that the vectors y µ have the maximum possible information content about the high-dimensional ξµ. Equivalently, one can show that, for fixed dimensionality M, PCA can be interpreted as a linear autoencoder, cf. Sec. 5.5.1, realizing a dimensionality reducing a linear mapping and back-transformation: y µ k = uk · ξµ, ξµ est = M k=1 yµ k uk, (8.15) realizes the smallest quadratic reconstruction error (5.37). Introducing the ma- trix U = [u1, u2, . . . uM]⊤∈RM×N we can write (8.15) in the compact form yµ = U ξµ, ξµ est = U ⊤yµ. (8.16) Generically, we will apply PCA to high-dimensional data sets, in particular if the number of examples is lower than the nominal dimension, i.e. P < N. | 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 leading eigenvectors carry most information, the mapping is truncated at relatively small M, neglecting variations along minor eigendirections as noise. Powerful tools exist that can be used to determine the leading eigenvectors of the symmetric, semi-definite matrix C. Often, numerical methods for the more general singular value decomposition (SVD) are preferred [Str19,DFO20]. SVD reduces to eigenvalue decomposition for diagonalizable matrices, but it is claimed to be numerically more stable. Whitening PCA can also be used for a so-called Whitening transformation. The trans- formed coordinates yµ k = uk · ξµ √λk (8.17) display zero mean and unit variance for all k. Hence, on the level of second order statistics, the data appears totally isotropic with an identity covariance matrix. Note that this does not imply that there is no structure left in the data. Whitening can be useful when analysing properties of the data which concern higher order statistics as in Independent Component Analysis, see Sec. 8.3.3. 8.3.2 PCA by Hebbian learning It is instructive to study a very simple numerical procedure to compute u1, which corresponds to the power method or von Mises iteration [MP29,Wil65]. As we will see, it relates to Hebbian learning and can be extended to the computation of several leading eigenvectors from a neural network perspective. Consider an initial vector w(0) which can be expanded in terms of the eigen- vectors ui of C: w(0) = N j=1 aj uj with coefficients aj ∈R and a1 ∕= 0. Repeated multiplication (from the left) with C leads to the t-th iterate w(t) = Ctw(0) = N j=1 aj λt j uj = λt 1 a1u1 + N j=2 λj λ1 t aj uj ≈ λt 1a1u1 for t →∞, (8.18) where we omit contributions that vanish like O λ2 λ1 t . | 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 modified iteration by replacing C by the shifted matrix [IN + η C] with η > 0 where IN is the N-dim. identity. This matrix has the same eigenvectors as C and ordered eigenvalues 1 + ηλj. The corresponding iteration reads w(t) = [IN + ηC] w(t−1) = w(t−1)+ η Cw(t−1) = w(t−1)+ η 1 P P µ=1 yµ(t) ξµ (8.19) where yµ(t) = w(t −1) · ξµ and we exploit that for general w ∈RN [Cw]i = N j=1 Cijwj = 1 P P µ=1 ξµ i N j=1 ξµ j wj = 1 P P µ=1 ξµ i yµ. Instead of computing the sum over µ = 1, . . . P in each iteration step (8.19) we can update the vector w in single steps of the form w(τ) = w(τ −1) + η yµ(τ) ξµ (8.20) where w(0) is the initial vector and the index µ on the r.h.s follows the sequence µ = 1, 2 . . . P, 1, 2, . . . which corresponds to the repeated presentation of all data in a fixed sequential order 5. Note that the update (8.20) can be interpreted as Hebbian Learning in a linear neuron with inputs ξj and output y = w · ξ. The update can also be derived as the gradient based maximization of cost function Evar in Eq. (8.9). In practice, it makes sense to normalize the weight vector after each update step in order to avoid numerical problems of diverging or vanishing |w|: w(τ) = w(τ −1) + η yµ(τ) ξτ |w(τ −1) + η yµ(τ) ξτ| (8.21) with yµ(τ) = w(τ −1) · ξµ. Assuming that the previous w(τ −1) was already normalized, we note that |w(τ −1) + η yµ(τ) ξτ| = 1 + 2η[yµ(τ)]2 + O(η2). The last term in the square root can be neglected for small learning rates η →0. In the same limit (1 + ηz)−1/2 ≈(1 + ηz/2) and we therefore obtain Oja’s rule w(τ) = w(τ −1) + η yµ(τ) ξτ −[yµ(τ)]2w(τ −1) . (8.22) 5The change from Eq. (8.19) to (8.20) is reminiscent of the difference between Gauss-Seidel and Jacobi iterations for linear systems [Fle00], see Sec. 3.7.2 It also resembles the relation between batch and stochastic gradient descent, cf. see Appendix A.48. | 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 principle, one could apply the power method also for the second and all following principal components. Theoretically, we could avoid contributions of u1 in the iteration (8.18) by starting from a w(0) with coefficient a1 = 0. Analogously, we would set a1 = a2 = . . . am−1 = 0 when computing um. Similarly, one could consider the modified matrices Cm = C − m−1 k=1 λk[uku⊤ k ] for the iteration of the vector wm. However, both ideas are at risk to fail in practice, as numerical inaccuracies will always introduce small but non-zero contributions of u1 which will blow up in the iterations, if not corrected for. It is more promising to iterate a set of vectors {wj}M j=1 in parallel and im- pose appropriate conditions after each step, e.g. by means of a Gram-Schmidt orthonormalization. Two closely related extensions of Oja’s Rule are based on this concept. We present here only the single step variants and refer the reader to the original literature [Oja89,San89]. For details and convergence proofs, see also [HKP91] for a more elaborate discussion. Oja’s subspace algorithm and Sanger’s rule (8.23) Initialize a random wm(0) with |wm(0)| = 1 for m = 1, 2, . . . M At discrete time step τ - determine the index µ of the current example (sequential presentation) - update the vectors wm(τ), m = 1, 2 . . . M according to Oja’s subspace algorithm: wm(τ) = wm(τ −1) + η yµ m(τ) ξµ − M k=1 yµ k(τ)wk(τ −1) (8.24) or according to Sanger’s rule: wm(τ) = wm(τ −1) + η yµ m(τ) ξµ − m k=1 yµ k(τ)wk(τ −1) (8.25) where yµ k(τ) = wk(τ −1) · ξµ(τ). | 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 orthogonalization of vectors. Note that in Oja’s subspace algorithm the sum in Eq. (8.24) is over all indices k = 1, 2, . . . M, while in Sanger’s rule (8.25) it is truncated to k = 1, 2, . . . m for the m-th vector. This imposes a hierarchy among the iterated weight vec- tors. In fact, one can show that Sanger’s algorithm yields the ordered principal components, i.e. wk(τ) →uk for large τ. On the contrary, in Oja’s subspace algorithm the vectors {wk}M k=1 converge to form an arbitrary orthonormal basis of the same subspace. They do not necessarily become identical with uk and do not display the same order with respect to the partial variances of the data. Orthogonal vectors wk ⊥wl result in uncorrelated projections yµ k and yµ l . In fact, the updates (8.25) and (8.24) can also be derived as (single example) gradient maximization of cost function (8.9) complemented by penalty terms of the form −[yµ k yµ l ]2 for k ∕= l. 8.3.3 Independent Component Analysis As discussed above, PCA identifies directions in which uncorrelated projections display the largest variances. In Independent Component Analysis (ICA), the goal is to find orthogonal directions which are independent in a broader sense [HTF01, Sto04, HO00]. ICA is, for instance, used for Blind Source Separation to identify independent sources in a mixed signal [CJ10]. Algorithms based on Hebbian Learning have also been suggested for Independent Component Analysis, see e.g. [CLG08,LG19]. Typically, the problem is addressed by considering a proxy for statistical independence. Two popular concepts are a) Mutual information Here, projections are identified which carry as little information about each other as possible. For projections x = w⊤ξ and y = v = v⊤ξ the mutual information can be determined as the relative entropy [HTF01] Ix,y = x∈X y∈Y px,y(x, y) log px,y(x, y) px(x) py(y) with the joint density px,y in the domain X ×Y and the marginal densities px and py. Minimizing Ix,y identifies independent directions w and v in feature space. b) Deviation from Gaussianity Orthogonal directions in which the projections appear least Gaussian are expected to be most interesting and potentially relevant for the task at hand [HO00]. They would display, for instance, very skewed or multimodal histograms of projections. | 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 value) of the kurtosis of projections yµ = w · ξµ [HO00, HTF01]. It is defined as kurtosis = (y −y)4 (y −y)22 −3, (8.26) where (. . .) denotes means of the type y = 1 P P µ=1 yµ. Very often, the data is centered and whitened in a first step, Eq. (8.17), appearing isotropic with unit variance and zero mean in any direction. In this case, Eq. (8.26) simplifies to kurtosis = y4 −3. (8.27) The kurtosis as defined here is zero for normal densities.6 Projections yµ = w⊤ξ with maximum kurtosis (in absolute value) can serve as a proxy for directions in which the data differs most from Gaussianity. Figure 8.5 displays histograms of a unimodal density with kurtosis< 0 (left panel), which appears more bumpy than a Gaussian (center panel), and a density with kurtosis> 0, which appears pointier. Similarly, other non-Gaussian densities, e.g. multimodal or skewed ones, also have a non-zero kurtosis. 8.4 Clustering and Vector Quantization The identification and exploration of structures in a given data set can constitute a very useful preprocessing step. Quite often, methods of unsupervised learning are applied before the actual classification or regression scheme is implemented. The purpose can be to obtain insight into the complexity of the problem and into the difficulties that might be expected. If, for example, the data set contains several well-defined clusters or groups of similar feature vectors, this knowledge could be used in the design of a specific classifier. Similarly, density estimation techniques can be applied to obtain knowledge about the general statistical properties of the data. 6Note that the additive term −3 is sometimes omitted in the literature. Then, the kurtosis of a Gaussian density is obviously 3. | 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 Basic clustering methods A variety of specific clustering techniques exist. Many of them are based on suitable distance measures in feature space like the familiar Euclidean metrics in the simplest case. In methods of hierarchical clustering the distance D(Ca, Cb) between two clusters is derived from the pairwise d(xa, xb) of their individual elements xa ∈Ca and xb ∈Cb. Prominent criteria for the computation of cluster distances are known as ◦single linkage with D(Ca, Cb) = min {d(xa, xb)}xa∈Ca,xb∈Cb , where the closest pair of feature vectors determines D(Ca, Cb). ◦average linkage where D(Ca, Cb) = 1 |Ca||Cb| xa∈Ca xb∈Cb d(xa, xb) is determined by the average pair-wise distance. Here, the number of vectors contained in a cluster C is denoted as |C|. ◦complete linkage with D(Ca, Cb) = max {d(xa, xb)}xa∈Ca,xb∈Cb corresponds to the largest pairwise distance of vectors in Ca and Cb. These and similar measures can be used in so-called hierarchical clustering ap- proaches like ◦agglomerative clustering (bottom up): Here, every data point is initially considered a cluster. In every step the two clusters Ca and Cb, which are the closest according to some criterion D(Ca, Cb), are merged into one cluster. ◦divisive clustering (top down): Here one starts by assigning all of the data to one cluster. In every step, one of the current clusters is selected and split into two sub-clusters. Var- ious criteria can be considered in the cluster selection and to guide the actual division, see e.g. [DHS00]. Unlike K-means or similar methods, hierarchical clustering does not require a predefined number of clusters. On the contrary, the procedure generates a hierarchical tree of clusters. Deeper levels of this so-called dendrogram represent splits of the data set into an increasing number of sub-clusters. A particular configuration can be chosen once the dendrogram is constructed. | 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-called prototypes should reflect the frequency of observations in feature space. The goal could be to reduce storage needs or to reveal structures such as clusters in the data. In the following we present a basic scheme for unsupervised Vector Quanti- zation by competitive learning [Koh97, HKP91, Bis95a, HTF01, BHV16]. Tech- nically, it resembles the methods of (supervised) Learning Vector Quantization discussed in Sec. 6.1 in that it also employs the concept of competitive learning. However, unsupervised VQ is applied to unlabeled feature vectors and its aim is the faithful representation of data sets, not an actual classification or regression task. Like most unsupervised learning techniques, VQ is also guided by a cost function. Here, it is optimized in terms of the prototype vectors. The objective function measures the quality of a particular prototype configuration with re- spect to a given set of vectors P = xµ ∈RNP µ=1. In neuroscience jargon, the input vectors can be interpreted as stimuli which activate the neurons or units that are characterized by N-dim. vectors w1, w2, . . . wK . The prototype vec- tors wk ∈RN can be seen as weight vectors, exemplars or expected stimuli, see [BHV16]. A very popular approach to VQ is based on the crisp assignment of any data point to the closest prototype, the so-called winner in terms of a pre-defined, fixed distance measure. We restrict the discussion to the use of simple squared Euclidean distance d(x, y) = (x −y)2 /2 for x, y ∈RN. Generalizations to alternative measures are in principle possible along the lines discussed in 6.1. A corresponding, suitable cost function is given by the so-called quantization error [Bis95a,Bis06,HTF01,BHV16]: HV Q = P µ=1 1 2 wk(µ) −xµ2 . (8.28) Here, wk(µ) ∈RN denotes the winning prototype with the smallest Euclidean distance from xµ ∈RN: d wk(µ), xµ ≤d wj, xµ for all j = 1, 2, . . . , K, (8.29) where ties are broken arbitrarily. Hence, HV Q accumulates the quadratic dis- tances of all feature vectors from their closest prototype. In this sense, the quantization error indicates how faithfully the set of feature vectors is repre- sented by the prototypes. Standard competitive VQ corresponds to the stochastic gradient descent based minimization of HV Q, see Appendix A.48. The resulting algorithm is very intuitive and could be obtained on purely heuristic grounds: at each discrete time step, a single randomly selected feature vector xµ is drawn with equal probability from the data set. Then, the currently closest prototype wk(µ) is determined according to Eq. (8.29). Similar to the supervised LVQ1 algorithm | 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 (Vector Quantization) – at time step t, select a single feature vector xµ randomly from the data set P with equal probability 1/P – identify the winning prototype, i.e. wk(µ) with d(wk(µ), xµ) = min d(wj, xµ) – update only the winning prototype according to: wk(µ)(t+1) = wk(µ)(t) + η(t) xµ −wk(µ)(t) . (8.30) The term competitive learning has been coined for this and several related training schemes as prototypes compete for updates [Koh97,Bis95a,Bis06,RMS92, BHV16]. The algorithm described above is an example of a Winner-Takes-All (WTA) scheme, extensions to the update of several prototypes at every time step have been considered also in the context of unsupervised Vector Quantization. As in any stochastic gradient descent, convergence of the prototype vectors has to be guaranteed by employing a time-dependent or adaptive learning rate η(t) which slowly approaches zero in the course of training, see the discussion of SGD in Appendix A.5. Competitive Vector Quantization is closely related to the well-known Lloyd’s algorithm or K-means algorithm [HTF01, Llo82, DHS00]. In contrast to the prescription (8.30) it considers all available data in each update of the system and alters all prototypes at a time: K-means algorithm (Lloyd’s algorithm) Given a data set xµ ∈RN (µ = 1, 2 . . . P), initialize K prototype vectors wj ∈RN (j = 1, 2, . . . , N). Repeat the following steps: A) assign every data point xµ to the nearest prototype wk(µ) with d wk(µ), xµ = min d(wj, xµ) . B) compute updated prototypes (centers) wj as the mean of all data points which were assigned to wj in (A): wj = P µ=1 xµδj,k(µ) P µ=1 δj,k(µ). (8.31) | 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 decrease with increasing K. Obviously, placing one prototype on each individual data point always gives the lowest possible quantization error HV Q = 0. A key difficulty of competitive VQ and the K-means algorithm is that the objective function HV Q can display many sub-optimal local minima. Among other problems, this can lead to a strong dependence of the training outcome on the initial prototype positions. As an extreme example, placing a prototype in an empty region of feature space can prevent it from ever being identified as the winner for any of the data points, thus leaving it unchanged in a WTA training process. The term dead unit has been coined for such a prototype. For competitive learning, rank-based schemes have been suggested which help to overcome this problem by updating not only the winner. Instead, pro- totypes are ranked according to their proximity to the presented feature vector, with e.g. the rank r = 1 of the winner, r = 2 of the second closest prototype etc. Generally, the magnitude of the update of a prototype is a decreasing function of r. A prominent example for the application of rank-based updates is the so- called Neural Gas algorithm [MBS93,RMS92,BHV16], which employs relatively large numbers of prototypes representing the density of data in feature space. The idea has also been extended in the context of supervised learning [HSV05]. In Self-Organizing Maps (SOM), competitive learning is also the key ingre- dient [Koh97]. In addition, prototypes are associated with a low-dimensional latent space, in which they form a grid. Updates affect the winning prototype as defined in conventional VQ, but neighbors of the winner in the grid are also updated. This way, the SOM can approximately preserve and visualize topolo- gies, i.e. neighborhood relations, in the latent space. For more information on this popular method see e.g. [Koh97,RMS92,BHV16]. Vector Quantization vs. Clustering Frequently, Vector Quantization is confused or even identified with clustering as indicated by the frequent use of the suggestive term K-means clustering in the literature. This is based on the idea that prototypes or centers should always represent pronounced clusters of similar data points. Then, the quantization error would correspond to the average within-cluster distances. However, it is important to note that VQ is well-defined and useful even if there are no clusters present in the data. Figure 8.6 shows a few illustra- tive situations, where prototypes represent simple, two-dimensional data with minimal HV Q. Panel (a) displays a single, elongated cluster represented by prototypes which characterize the variation of observed feature vectors. Here, the minimization of HV Q does not correspond to the identification of a set of clusters. Panel (b) shows an idealized case of approximately spherical clusters. Each of the two clusters can be represented by one prototype. In panel (c) of | 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 subpanels are referred to in the text of Sec. 8.4.3. Fig. 8.6, clusters appear to be elongated along one of the axes in feature space. A minimal value of the quantization error can be achieved by placing the pro- totypes in the space between the apparent clusters, where hardly any examples are observed. Panel (c) illustrates the fact that the outcome of VQ can be highly sensitive to coordinate transformations, even if they are linear. Panel (d) of the Fig. 8.6 displays a situation in which a smaller, separated cluster is not identified at all. Although these few data points contribute large distances, their total contribution do not have a significant influence on the position of the prototypes when minimizing HV Q. The number of prototypes or clusters As discussed in Sec. 2, unsupervised learning including clustering and Vector Quantization frequently lack simple performance measures which makes it dif- ficult to select a model of suitable complexity. The goal of Vector Quantization can be formulated mathematically in terms of HV Q, but the cost function is strictly speaking only suitable for comparing systems which have the same com- plexity, i.e. the same number of prototypes. Similar restrictions apply to explicit | 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 of additional information like domain knowledge providing a ground truth, it is impossible to infer the correct or optimal number of clus- ters or prototypes from the data set P alone. In this sense, clustering and VQ are ill-defined problems, ultimately their evaluation depends on the user’s preferences and subjective quality criteria. Several heuristic ideas have been suggested for the determination of a suit- able number of clusters or prototypes. Here we illustrate the popular elbow method [Tho53] in terms of an application of the K-means algorithm. Fig. 8.7 displays the quantization error HV Q as obtained in the Iris flower data set [Fis36] by applying the algorithm with different choices of K. In this simple example, we can conclude that K = 3 appears to be a reasonable choice. For K = 1, 2 the quantization is much higher, while for K > 3 no further, significant reduction of HV Q is achieved. The curve displays a pronounced elbow shape that suggests K = 3, hence the term elbow method.7 Of course, the insight does not come as a surprise as the data set was al- ready considered in Sec. 6.2.2 in the context of supervised learning. It contains samples from three different classes corresponding to a more or less pronounced cluster structure, see Fig. 6.2 for a discriminative visualization. In more gen- eral practical situations, the shape of the corresponding curves and the elbow is frequently much less conclusive. Several related and alternative methods have been suggested, some of which are based on more rigorous statistical arguments. As one example, R. Tibshirani et al. suggested the so-called gap statistics as a criterion for the estimation of the number of clusters in K-means or other methods [TWH02]. 7Depending on the actual quality criterion and the visual representation, the elbow could also be a knee. | 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 overviews of relevant methods and theoretical background. Several basic concepts can be applied in density estimation. In paramet- ric methods, a particular statistical model is postulated and adapted. More precisely, a particular functional form of the density is assumed. Then, the optimization of its parameters can be formulated as an unsupervised learning process based on the observed feature vectors in P. So-called non-parametric methods 8 do not assume a specific functional form a priori, but aim to infer a descriptive density directly from the data [Bis95a]. Examples are so-called histogram methods or approaches using local kernels (not to be confused with kernel functions in the context of SVM and related methods). Here we focus on the important family of mixture models, which are some- times referred to as semi-parametric [Bis95a]. We first discuss some basic ideas behind parametric density estimation and then present the particular popular example of Gaussian Mixture Models (GMM), see e.g. [Bis95a,HTF01,DFO20]. 8.5.1 Parametric density estimation In the parametric approach we make explicit assumptions about a probability density that could explain the observed data set. Very frequently we will assume that feature vectors ξµ ∈RN can be interpreted as generated independently according to a specific identical N-dim. density. We furthermore assume an appropriate parametric form and, hence, we can write the likelihood and log- likelihood of the observed data as L(Θ) = P µ=1 p(ξµ|Θ) and ℓ(Θ) = ln(L(Θ)) = P µ=1 ln p(ξµ|Θ). (8.32) Here, the vector Θ ∈RK concatenates the K parameters of the model density p(ξ|Θ). The dimension K of Θ as well as the type of parameters depend on the actual structure of the considered model. The optimization of ℓ(Θ) yields the corresponding maximum-likelihood model. Similar to the discussion in Sec. 2.13 we can extend the formalism by introduc- ing a prior density po(Θ) and derive the resulting maximum a posteriori (MAP) model. Furthermore, Bayesian estimation techniques can be applied to obtain a probabilistic description of the model parameters, see Sec. 2.13 for a discussion 8Despite the suggestive name, non-parametric methods may very well comprise parameters and hyper-parameters. | 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 also in the context of density estimation. The assumed model density should be appropriate to represent the complexity of the data. While the formalism is very powerful, it can be difficult to define suitable models which allow for an analytical or efficient numerical treatment. A particularly successful concept is the consideration of adaptive models which are defined as mixtures of specific basis functions. Gaussian densities p(ξ|Θ) constitute a particular popular and convenient choice. Here we restrict our analysis to the particularly simple case with p(ξ|Θ) = M m=1 pm p(ξ|m, σm, wm) (8.33) with p(ξ|m, σm, wm) = 2πσ2 m −N/2 exp −1 2σ2m (ξ −wm)2 which also factorizes with respects to the components ξj. For simplicity we as- sume here that the contributing densities are isotropic with covariance matrices σM IN. Extensions to more general N-dimensional Gaussian densities are of course possible. The model parameters Θ in (8.33) are given by the union of the sets σm ∈R, wm ∈RN, pm ∈R for m = 1, 2, . . . M. Hence, Θ ∈RK with K = M(N + 2) in this special case. Note that the so-called mixing parameters pm quantify the contribution of the individual Gaussians and have to satisfy the conditions 0 ≤pm ≤1 for all m and M m=1 pm = 1. (8.34) Given a particular realization of the model we assign a feature vector ξ to one of the contributing Gaussians with probability Qm = pm p(ξ|m, σm, wm) M k=1 pk p(ξ|k, σk, wk) = σ−N m pm exp − 1 2σ2 m (ξ −wm)2 M k=1 σ−N k pk exp − 1 2σ2 k (ξ −wk)2. (8.35) Note that also the Qm are properly normalized with M m=1 Qm = 1. Analo- gously, we define the quantities Qµ m by (8.35) evaluated in ξ = ξµ. For the specific model density (8.33), the log-likelihood (8.32) reads ℓ(Θ) = P µ=1 ln M k=1 pkσ−N k exp −1 2σ2 k (ξµ −wm)2 + const. (8.36) | 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 the assignment probabilities Qµ m defined in Eq. (8.35). Note that for the derivation of (8.39) the normalization m pm = 1 has to be taken into account explicitly [Bis95a]. The equations can be solved self-consistently by means of the intuitive natural iteration method, see e.g. [Kik76]. If we interpret the r.h.s. as to define the next iterates of the quantities on the l.h.s. we obtain the following algorithm: Gaussian Mixture Model (isotropic Gaussian contributions) - initialize wm, σm, pm for m = 1, 2, . . . , M - update the model parameters according the following iterative procedure: wm(t + 1) = P µ=1 Qµ m(t) Qµ m(t) P ν=1 Qνm(t) ξµ (8.40) σ2 m(t + 1) = P µ=1 Qµ m(t) P ν=1 Qνm(t) ξµ −wm(t + 1) 2 (8.41) pm(t + 1) = 1 P P µ=1 Qµ m(t). (8.42) Note that here wm(t + 1) appears on the r.h.s. of (8.41), but the terms Qµ m(t) are evaluated by inserting the previous wm(t) into Eq. (8.35). While this de- tail is not essential for the iteration to work, it is very interesting to note that this specific form can be derived as an Expectation-Maximization (EM) proce- dure for the optimization of the (log-)likelihood [Bis95a,HTF01,DFO20]. This methodological framework has been formalized by Dempster et al. for quite general problems involving incomplete data, see [DLR77]. In our density esti- mation problem we can interpret the assignment probabilities Qµ m as unknown latent variables. Detailed discussions of the very versatile EM-approach can be found in textbooks like [Bis95a] or [HTF01]. Figure 8.8 illustrates the application of the GMM algorithm in terms of a rather simple multi-modal density of two-dimensional data points (left panel). | 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 after 20 EM-steps according to (8.40-8.42). A mixture of six Gaussians is adapted to the data following the iteration (8.40- 8.42), the center panel of Fig. 8.8 displays the initial configuration of the system with pk = 1/6. After as few as 20 update steps, the density is approximated very well. Note that the model is overly complex with 6 Gaussians representing only 4 clusters in the data. However, this does not appear to constitute a problem in this simple setting. The complexity is reduced by coinciding Gaussians with equal means and variances or by eliminating redundant contributions by having mixing parameters pk →0. The actual realization of the estimated density in terms of the model parameters will depend on the initialization in this example. It is very instructive to consider the algorithm (8.40–8.42) in an extremely simplifying limit. Assume that the variances are equal and fixed in all compo- nents of the GMM, i.e. σ2 m = σ2 which simplifies Eq. (8.35): Qm = pm exp − 1 2σ2 (ξ −wm)2 M k=1 pk exp − 1 2σ2 (ξ −wk)2. (8.43) If we now take the limit σ →0, the sum is dominated by the largest summand (the term with the smallest (ξ −wk)2), which is exponentially larger than all other ones. Consequently we obtain lim σ→0 Qm = 1 if (ξ −wm)2 ≤(ξ −wk)2 for all k ∕= m 0 else. (8.44) Hence, in this limit, data points are assigned in a crisp way to the closest wm according to Euclidean distance. In the GMM algorithm (8.40–8.42), the updates of the variances become obsolete and the pm are simply computed as the fraction of crisp assignments to wm(t), while the new wm(t+1) are obtained as the means of the data points currently assigned to wm(t). | 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 of more general statistical modelling schemes. 8.6 Missing values and imputation techniques Many real world data sets are compromised by missing values, i.e. components of feature vectors that have not been observed, registered or communicated properly. Missing values can occur due to a variety of more or less complex reasons, see [Gho21] (chapter 3) and [GBB+23] for a comprehensive discussion. Quite frequently, the use of a certain spreadsheet based software tool leads to unwanted missingness or other artefacts and makes it difficult to excel. These are not explicitly discussed here, but always should be taken into consideration as a potential source of error. In the literature, the following rather coarse categorization of missingness can be found [Lit88, GLSGFV10, LR02, Gho21, GBB+23], examples are taken from [Bla15]9 ◦MCAR: missing completely at random Missingness is considered to be completely at random if the absence or presence of a feature does not depend on its value or on the value and/or missingness of other features. As an example, an accidentally damaged blood sample in a medical study would result in missing the observation of a particular feature. ◦MAR: missing at random In this case, the fact that a feature is missing might be predictable from other information, but does not depend on the potential value of the miss- ing feature. As an example, similar to the one presented in [Bla15], a person may miss an IQ test because they are ill on the day that the IQ test is given. This missingness relates to other available information about the person, but does not depend on the potential outcome of the test, i.e. the feature value itself. ◦MNAR: missing not at random If missingness is specifically related to the feature that is missing, the somewhat vague term not at random is used. For instance, a person might have avoided a drug test resulting in a missing observation, because they took drugs and the outcome would be positive. Another example would be a feature that cannot be registered whenever it exceeds an allowed range of values, e.g. due to technical limitations of the measurement process. The distinction of these types of missingness is not always very clear. More- over, it can be difficult (if not impossible) to infer the underlying reason for 9See also https://www-users.york.ac.uk/~mb55/intro/typemiss4.htm | 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 properly into account. Consequently, the methods discussed in the following are most appropriate for data affected by MCAR or MAR type missingness. 8.6.1 Approaches without explicit imputation Depending on the frequency and nature of missing features, simple strategies can be applied which ignore or eliminate the missingness beforehand or as an integral part of the training process. ◦Deletion The most straightforward idea to handle missingness is to include only complete feature vectors in the analysis and omit all others. This is appealing since it does not require explicit manipulations of the data. However, disregarding a subset of samples might introduce biases. In any case, simple deletion is only feasible if enough training data are available to begin with. In addition, one would have to reject all incomplete data in validation, test or working phase. A similar (also problematic) strategy is to omit features entirely if they appear to be missing in a significant fraction of the available samples. ◦Training algorithms that can handle missingness In some machine learning frameworks it is possible to restrict their application to the features that are present in a given observation. As an example, the computation and ranking of pairwise distances in a subset of samples can be re- stricted to the set of features that is present in all involved feature vectors. The concept could be applied, for instance, in Nearest Neighbor Classification. Sim- ilarly, in LVQ and other prototype based systems, cf. Chapter 6, the distances of an incomplete test or training sample from all prototypes can be computed and compared (excluding the missing components) for the identification of the closest prototype, see e.g. [GBV+17]. Another example of an algorithm that can handle missingness (and noise) without explicit imputation is the so-called Probabilistic Random Forest [RBS19]. 8.6.2 Imputation based on available data The most widely used approach to handle missing values is imputation, i.e. the replacement of missing features by more or less sophisticated estimates based on the available data. Several strategies and modifications of the basic idea are explained in the following. For details and references, see e.g. [Gho21,GBB+23]. ◦Naive estimates A straightforward but naive idea is to replace a missing value in, say, feature | 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 used for imputation in a set of labeled training data. But this estimate would be unavailable for unlabeled feature vectors in the test or working phase. ◦Nearest-Neighbor imputation Applying a Nearest-Neighbor (NN) (or K-NN) approach partly solves the above mentioned problem and constitutes a popular tool for imputation. Given an incomplete feature vector ξ with missing value ξj, one can determine the nearest neighbor(s) of ξ in the training set e.g. in terms of the partial Euclidean distance w.r.t. the available dimensions. Then, the value of ξj in the nearest neighbor or an appropriate estimate based on the K nearest neighbors can be imputed. ◦Regression based imputation If a feature ξj can be expected to be correlated with other features ξk with k ∕= j, imputation can be based on a regression scheme. From the available data we can infer this dependency by means of linear or non-linear regression and obtain a prediction for the missing value. ◦Cold deck and hot deck imputation The term cold deck imputation has been coined for situations in which a sepa- rate data set is used for imputing missing values in the training data or when testing/applying the trained system. On the contrary, in hot deck settings, the available (training) data set itself is used for determining the imputed values. ◦Generative modelling for imputation Methods discussed in Sec. 8.4, can be employed to estimate the density of data or a particular feature from a given data set. If the occurrence of feature ξj is modelled by e.g. a mixture of Gaussians, the resulting GMM can be used to generate random values for the imputation of missing values accordingly. ◦Multiple imputation Frequently, more than one imputation value is generated per missing feature. Generative approaches or randomized versions of regression based imputation can be used to provide multiple versions of the imputed data set. Then, training and validation can be performed on a number of versions in order to obtain reliable performance estimates. As an example, the so-called Multiple Imputation by Chained Equations (MICE) has recently become popular [ASFL11, Gho21, GBB+23]. There, in a first step all missing values are replaced by simple mean or median impu- tation, with the exception of one selected feature dimension, say, ξk. Next, a regression based imputation is used to replace the missing values of ξk in the data set. Subsequently the missing values of a different feature ξj(j ∕= k) are | 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 varying the order of features, thus obtaining several versions of the imputed data set. 8.7 Over- and undersampling, augmentation Here we discuss the problems that arise when a classification problem is im- balanced in the sense that the prevalences of the individual classes are very different. Similar difficulties occur in regression problems with skewed and/or multimodal distributions of the target variable, but here we restrict the discus- sion to classification. For reviews of imbalance related strategies and further references consult, for instance [Cha09,HG09]. Class-imbalance can be handled in various ways when validating a given classifier, see the discussion in Sec. 7.4. However, it is often necessary to take the imbalance into account in the training phase already. Strong over-representation of certain classes in the training data can lead to very poor performance with respect to minority classes. Note that an imbalanced training set does not necessarily reflect the prevalences we can expect in the working phase. 8.7.1 Weighted cost functions Assume that a training set comprises a total of P = C j=1 Pj examples, with Pj of those representing class j in a C-class problem. Virtually all objective functions we have considered for training are of the general form E(W) = 1 P P µ=1 eµ (8.45) with the contribution eµ of a single example in the data set. We can balance the influence of the different classes by considering the weighted cost function Ebal(W) = C j=1 1 Pj Pj µj=1 eµj (8.46) where the partial sums over µj = 1, 2, . . . Pj contain only examples from class j ∈{1, 2, . . . C}. Note that a gradient descent optimization of Ebal leads to updates that are equivalent to descent in the original E, Eq. (8.45), with class- specific learning rates ηj ∝1/Pj. 8.7.2 Undersampling Another straightforward way to compensate for class imbalances is to per- form the actual training on sets with balanced class composition. This can | 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 remain unbalanced, but the evaluation criteria should be robust against imbalance, like the BAC or other measures discussed in Sec. 7.4.2. The random selection could be done once, disregarding the remaining data. However, this strategy would not make use of all available information and bears the risk of selecting atypical cases. Instead, the random undersampling should be performed repeatedly, which enables the computation of averaged performance measures. 8.7.3 Oversampling One can also aim at increasing the effective influence of the underrepresented minority classes by generating additional examples for training. The two main ideas are described in the following. Random oversampling In this simple approach, the training set is augmented by exact copies of ran- domly selected examples in the underrepresented classes. It can be realized by random selection of examples with replacement. For loss functions of the general form (8.45), the effect of including copied examples in the training data is very similar to weighting the classes as in Eq. (8.46). However, in the corrected cost function every example from a given class has the same weight. In practice, random oversampling in the underrepresented classes and un- dersampling of the overrepresented ones are often combined. Generative oversampling Instead of using exact copies of available examples, one can aim at generating synthetic data which reflect the statistical properties of the minority classes. The simplest idea would be to augment the training set by noisy copies of randomly selected feature vectors. In more sophisticated approaches one per- forms a suitable density estimation and uses the resulting model for generating additional samples. The Synthetic Minority Oversampling Technique (SMOTE) [CBHK02] fol- lows a similar, yet simpler concept. The basic scheme is summarized in the following: | 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 point at a random position on the line connecting the two selected feature vectors, assign it to the same class. Several modifications and extensions can be considered. For example, the neigh- bor identification and construction of the synthetic data point could be based on different distance measures, see [GBV+17] for an example of applying geodesic SMOTE in angle-based classification. Practical issues The suitability of the approaches discussed in Sec. 8.7.1–8.7.3 clearly depends on the availability of example data from all classes and on the details of the problem at hand. Both the weighting of classes in the cost function and undersampling are easy to realize since they do not rely on generating synthetic data. Clearly, undersampling is only feasible for data sets with a reasonably large number of examples for each class to begin with. However, if that is not the case, the application of machine learning is questionable anyway. Generative oversampling cannot really create fundamentally novel informa- tion, since the augmented feature vectors more or less faithfully reflect the properties of the original data set. As a consequence, the oversampling and subsequent training bears the risk of overfitting to the available minority class data. In addition, generative oversampling introduces problems of model and parameter selection, e.g. concerning the number of neighbors in SMOTE or the choice of a model density to begin with. Imbalanced data can constitute a significant challenge in practical applica- tions of machine learning. Measures to compensate the imbalance should be applied with care and they should always be evaluated in a proper validation process. 8.7.4 Data augmentation In practical applications, even balanced data sets are often augmented by addi- tional, artificial training data. Data augmentation can be naively motivated as a way to improve the training by providing more example data. In principle, one can employ all methods discussed for oversampling in the previous section also for the generation of additional data. The concept of data augmentation | 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 generate genuinely novel infor- mation from a given data set alone. However, a few cases in which augmentation appears justifiable can be identified: ◦domain knowledge based models In specific applications, generative models of the expected data in the working phase may be available, which are based on explicit domain knowl- edge. Such models can be used to generate surrogate data for training and testing. Assume that the characteristics of a novel instrument, say a tele- scope in astronomy, are known in detail. If, in addition, key properties of the objects of interest are known, artificial data can be generated by means of simulations of the observation process and subsequently used to (pre-) train a classifier or regression system, see [RMBJ21] for just one recent example. Of course, systems trained on surrogate data should eventually be validated and tested in a real world setting. ◦increased robustness against noise The robustness of a classifier w.r.t. input noise can be increased by com- plementing the training set by noisy copies of the original data. In fact, this strategy can be viewed as a regularization technique [Bis95b,GBC16]. Obviously the noise must not be too strong and should reflect the expected level in real data. ◦imposing invariances Analogous strategies can be applied in order to achieve robustness with respect to other expected variations in feature space. This plays an impor- tant role in image processing tasks. The original data is often subjected to systematic variations, such as rotation, scaling, or skewing of objects. A properly designed training set will result in a classifier or regression system that displays the desired invariances. It is essential that the applied variations are suitable for the task at hand: while e.g. galaxy classification [NWB+19] will not be affected by arbitrary rotations in the image plane, as all orientations should occur in the data anway. On the contrary, handwritten character recognition is insensitive to rotations of the symbols only in a very small range of angles. In many cases, data set augmentation appears to be a cheap but efficient way to increase the performance of the trained system. However, systematic alterna- tives should always be considered. In particular, invariances could be imposed through preprocessing, i.e. the extraction of invariant features. Even more ele- gantly, they can be achieved by appropriate network design and modified loss functions which realize invariant functions, see [WBJH18] for an example and further references. | 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 |
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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 strategies. We also discuss basic aspects of constrained optimization. Note that we refrain from providing the precise (yet usually mild) math- ematical conditions for the validity of expansions and applicability of theo- rems. For instance, we assume implicitly that all considered functions are continuous and differentiable, that second derivatives are symmetric, etc. For more rigorous presentations we refer the reader to the mathematical literature, e.g [Fle00,PP12,PAH19,Str19,DFO20]. A.1 Multi-dimensional Taylor expansion Consider a real-valued function of the form f : x ∈Rd →f(x) ∈R. (A.1) If the value of the function f(xo) and its derivatives are known in some point xo, we can perform a d-dimensional Taylor expansion to obtain the approximation f(xo + h) ≈f(xo) + h⊤∇f(xo) + 1 2 h⊤H(xo) h + O |h|3 (A.2) in the vicinity of xo. Here, we assume that the norm of the deviation h ∈Rd is small: |h| ≈0. The first term merely corresponds to the simple estimate f(x) ≈f(xo) close to the reference point. The second term takes into account 225 | 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 first (partial) derivatives of the function with respect to the coordinates x. With the formal vector ∇= ∂ ∂x1 , ∂ ∂x2 , . . . ∂ ∂xd ⊤ (A.3) we obtain the gradient of f in xo, i.e. the vector of partial derivatives ∇f(xo) = ∂f ∂x1 , ∂f ∂x2 , , . . . , ∂f ∂xd ⊤ x=xo and h⊤∇f(xo) = d j=1 hj ∂f ∂xj x=xo . (A.4) The third term in (A.2) involves the (d×d)-dimensional Hesse matrix or Hessian H(xo) of second derivatives with elements Hij(xo) = Hji(xo) = ∂2 ∂xi∂xj H x=xo and h⊤H(xo) h = d i,j=1 hiHij(xo)hj. (A.5) Higher order terms are obviously more involved than the quadratic approxi- mation (A.2). Here we assume implicitly, that a quadratic approximation is suitable for the extrema under consideration. | 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 be chosen anywhere in Rd without any restrictions. Hence, we do not have to consider minima at the border of allowed regions Note that the term local minimum is not synonymous with suboptimal mini- mum, it only implies that the function locally increases whenever moving away from the minimum. In this sense, a global minimum is also a local minimum. A.2.1 Necessary and sufficient conditions An obvious, necessary condition for the presence of a local minimum of f in the point x∗is that all first derivatives vanish: ∇f(x∗) = 0. (A.6) This follows immediately from the Taylor expansion (A.2) up to first order: If ∇f(x∗) ∕= 0, a small displacement with h = −η∇f(x∗) with η > 0 would result in f(x∗+ h) ≈f(x∗) −η|∇f(x)|2 < f(x∗) and, hence, x∗could not be a local minimum. Assuming that (A.6) is satisfied, we have with the shorthand H∗= H(x∗) f(x∗+ h) ≈ f(x∗) + 1 2h⊤H∗h (A.7) ∇f(x∗+ h) ≈ H∗h. (A.8) From Eq. (A.7) we obtain the sufficient condition x⊤H∗x > 0 for all x ∈Rd (A.9) in the presence of a local minimum in x∗. If this is the case, small steps away from x∗will always lead up-hill, increasing the function value. This is obviously only correct as long as |h| is small enough and the higher-order corrections O(|h|3) can be neglected. The Hessian H∗has to be positive definite, which is equivalent to the con- dition that all its eigenvalues are positive, i.e. H∗ui = ρ∗ i ui with ρ∗ i > 0 (i=1,2,. . . d) (A.10) Note that – due to the symmetry of H∗– all eigenvalues are real and we can construct an orthonormal set of eigenvectors {ui}d i=1 as a basis in Rd [PP12, Str19,DFO20]. | 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∗ i by comparison with the conventional, one-dimensional Taylor expansion with vanishing linear term. If the Hessian is indefinite, i.e. if it has positive and negative eigenvalues, the point x∗corresponds to a saddle point: in some of the eigendirections u∗ i it displays a local maximum, while in others it behaves like a one-dimensional local minimum, cf. Fig. A.1. Semi-definite H∗with ρ∗ i ≥0 (or ρ∗ i ≤0) require more careful considerations: Eigenvalues ρ∗ i = 0 can correspond either to an extended flat region (along ui) around a local extremum or to a saddle point in x∗. We refrain from discussing this subtlety in detail and refer the reader to e.g. [Fle00,Str19,DFO20]. A.2.2 Example: unsolvable systems of linear equations We consider a set of P linear equations in N variables w ∈RN of the form w⊤ξµ != yµ for µ = 1, 2, . . . , P. (A.11) For P > N the system is overdetermined and, in general, inconsistent, i.e. not solvable. Here, the notation is chosen to resemble the machine learning settings that we are considering throughout, where the coefficients and r.h.s. of the system are given by a data set of the familiar form D = ξµ ∈RN, yµ ∈R P µ=1 . Using the matrix and vector notation introduced in Sec. 2.2.2, we define Y = y1, y2, . . . yP ⊤∈RP and χ = ξ1, ξ2, . . . ξP ⊤ ∈RP ×N. (A.12) Now Eq. (A.11) is conveniently written as χ w != Y. (A.13) If χ happens to be an invertible square matrix with N =P, the solution of the problem is obviously w∗= χ−1 Y. However, if the set of equations is overdeter- mined and cannot be satisfied exactly, we can resort to approximative solutions. A natural and popular choice is to minimize the corresponding Sum of Squared Error (SSE), cf. Eq. (2.5), in the sense of a linear regression ESSE(w) = 1 2 P µ=1 w⊤ξµ −yµ2 = 1 2 [χw −Y ]2 = 1 2 w⊤ χ⊤χ w −w⊤χ⊤Y + 1 2 Y ⊤Y, (A.14) | 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 pseudoinverse [PP12,BH12] w∗= χ+ left Y with χ+ left = χ⊤χ −1 χ⊤ (A.16) under the condition that [χ⊤χ] is invertible. Here, the (N × P)-dim. matrix χ+ left satisfies χ+ leftχ = IN×N with the N-dim. identity. The assumption that [χ⊤χ] ∈RN×N is invertible is consistent with the assumption that the equations {w∗⊤ξµ = yµ}µ=1,2,...P cannot be satisfied si- multaneously, which - after all - was the motivation for minimizing ESSE. In our example we obtain the Hesse matrix of second derivatives in w∗as H∗ ik = ∂2ESSE ∂wi∂wk = P µ=1 ξµ i ξµ k or H∗= χ⊤χ ∈RN×N. (A.17) Note that for the quadratic ESSE a Taylor expansion up to second order would be exact in the minimum w∗, of course. The sufficient second order condition (A.9) corresponds to positive definite H∗. Here, it is guaranteed that H∗is at least positive semi-definite: u⊤χ⊤χu = [χu]2 ≥0 for any u ∈RN. In order to have positive definite H∗, the condition P ≥N must be fulfilled1. For P < N, the (N × N)-dim. matrix H∗= [χ⊤χ] cannot have full rank and is bound to have zero eigenvalues. Correspondingly, the system (A.11) has many solutions. We will address this case in the discussion of constrained optimization in the next section. Strict positive definiteness implies that H∗= [χ⊤χ] is invertible. We con- clude that – under this condition – the solution (A.16) exists and corresponds to a local minimum, which is also a global minimum since unrestricted quadratic costs cannot display other, local minima. In Sec. 2.2.2 we have briefly considered the case of singular [χ⊤χ] which cannot be inverted. There we solved the problem by introducing a non-singular [χ⊤χ + λIN] with λ > 0, which corresponds to a simple form of regularization. Alternatively, we can make use of the right pseudo-inverse which is introduced and discussed below in Sec. A.3.2. 1P ≥N is necessary, not sufficient. Correlations in the data set can still yield singular H∗. | 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: The constraints {gi(x)}n i=1 define the set of allowed argu- ments x. As a consequence, solutions of the problem (A.18) do not necessarily cor- respond to local minima of the (unrestricted) objective function f discussed in Sec. A.2. If it is possible to eliminate the conditions explicitly, one can transform the problem to an unconstrained one and proceed as before. Formally, constraints given in the form of equations as in (A.18) can be dealt with systematically by introducing Lagrange multipliers {λi ∈R}n i=1. We define the Lagrange function or Lagrangian L x, {λi}n i=1 = f(x) − n i=1 λi gi(x) (A.19) with the real-valued multipliers λi. Solutions of the constrained problem (A.18) satisfy the first order necessary conditions ∂L ∂xj ∗ = 0 d j=1 and ∂L ∂λi ∗ = 0. n i=1 (A.20) where we use the shorthand (...)|∗for the evaluation in x = x∗and {λi = λ∗ i } . The second set of conditions merely corresponds to the original constraints gi(x) = 0. Sufficient conditions for the presence of an optimum are non-trivial to formu- late in the presence of constraints, see [Fle00,PAH19] for a detailed discussion. Note that the extended (d + n) × (d + n)-dimensional Hessian of the Lagrange function L is indefinite, in general. The conditions (A.20) can often be exploited in order to eliminate some of the variables or, in fact, the multipliers and to simplify the structure of the optimization problem significantly. In Sec. 3.7.2 we present an important example of the above in terms of the Adaline problem, i.e. Widrow’s Adaptive Linear Neuron [WH60, WL90]. There, we consider the minimization of the norm |w|2 under linear equality constraints {w⊤ξµSµ T = 1}P µ=1. The actual perceptron weights can be eliminated by making use of the first order conditions, while the Lagrange multipliers play the role of the embedding strengths x ∈RP . The resulting optimization problem corresponds to an unconstrained maximization of the modified cost function (3.77). | 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. Consequently, for any given solution w of (A.21) we can construct a continuum of solutions w + io i=1 ci vi with arbitrary coefficients ci ∈R. The solution of minimal norm w∗is of particular interest. We can formulate the search for w∗as a quadratic optimization problem with linear equality constraints: Example: minimal norm solution of linear equations minimize w∈RN 1 2 w2 subject to χ w = Y. (A.22) Introducing multipliers λ ∈RP , we obtain the Lagrange function L w, λ = 1 2Nw2 −λ⊤(χw −Y ) (A.23) The first order necessary conditions (A.20) become w −χ⊤λ != 0 and χw −Y != 0. While the second condition is obvious, the first one suggests to eliminate w. We obtain the modified cost function L = −1 2 λ⊤[χχ⊤] + λ⊤Y and the stationarity condition −χχ⊤λ + Y != 0. Note that here [χχ⊤] ∈RP ×P is the counterpart of the (N ×N)-dim. matrix [χ⊤χ] that we have encountered earlier in Eqs. (A.14 ff). Here, we assume that P < N and that [χχ⊤] is invertible. We therefore get immediately λ = [χχ⊤]−1 Y ⇒w = χ⊤[χχ⊤]−1Y. This motivates the definition of the so-called right pseudoinverse [PP12,BH12] χ+ right = χ⊤[χχ⊤]−1 ⇒χ χ+ right = IP ×P , (A.24) with the P-dim. identity matrix IP ×P . | 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 exist even if χ⊤χ or χχ⊤is singular. The limits either coincide with the left pseudoinverse (2.8) or with χ+ right defined above. Note also that Eq. (A.25, left) corresponds to a limit of the regularization term (2.9) that we introduced heuristically in Sec. 2.2.2. A.3.3 Inequality constraints The concept of Lagrange multipliers has been extended to the presence of in- equality constraints in optimization problems of the form minimize x ∈Rd f(x) subject to n constraints {gi(x) ≥0}n i=1 . (A.26) We refrain from discussing the more general combination of inequality and equal- ity constraints, which leads to fairly obvious extensions of the following. For details we refer the reader to [Fle00,PAH19,Str19,DFO20]. Note that, in princi- ple, we could introduce pairs of inequality constraints gi(x) ≥0 and −gi(x) ≥0 simultaneously in order to include equality constraints, effectively. Formally, the corresponding Lagrange function (A.19) L x, {λi}n i=1 = f(x) − n i=1 λi gi(x) is the starting point. First order necessary conditions for a solution of (A.26) are given by the Kuhn-Tucker (KT) Theorem of optimization theory [Fle00,PAH19]. Here, they read Kuhn-Tucker conditions (only inequality constraints) ∇xL|∗= 0 ⇔ ∇xf|∗= n i=1 λ∗ i ∇xgi|∗ (stationarity) (A.27) gi(x∗) ≥0 for i = 1, 2, . . . n (constraints) (A.28) λ∗ i ≥0 for i = 1, 2, . . . n (non-neg. multipliers) (A.29) λ∗ i gi(x∗) = 0 for i = 1, 2, . . . n. (complementarity) (A.30) | 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 multipliers are non-negative in the op- timum. Individual multipliers λi > 0 correspond to so-called active constraints with gi(x) = 0, which follows from the complementarity condition (A.30). On the contrary, if gi(x) > 0 is satisfied with λi = 0, the constraint does not have to be enforced explicitly and is termed inactive. More detailed interpretations and discussions of the KT conditions can be found in the literature, see for instance [Fle00,PAH19]. Example: optimal stability in the perceptron As an important application of the KT theorem we exploit the necessary condi- tions (A.27–A.30) in the problem of maximum stability for the perceptron, see Sec. 3.7.3: minimize w ∈RN N 2 w2 subject to inequality constraints {Eµ ≥1}P µ=1 . The KT conditions are based on the Lagrangian L w,λ = N 2 w2 − P µ=1 λµ w⊤ξµSµ T −1 . which is the same as for the Adaline problem with equality constraints as given in Eq. (3.70). We work out the gradient ∇wL = N w − P µ=1 λµ ξµSµ T and obtain from (A.27-A.30) the following set of necessary conditions, which were already presented in Sec. 3.7: Kuhn-Tucker conditions (optimal stability) w∗ = 1 N P µ=1 λ∗µ ξµSµ T (embedding strengths λµ) (A.31) E∗µ = w∗⊤ξνSµ T ≥1 for all µ (linear separability) (A.32) λ∗µ ≥ 0 (not all λ∗µ = 0) (non-negative multipliers) (A.33) λ∗µ (E∗µ −1) = 0 for all µ (complementarity). (A.34) | 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 The concept of duality plays an important role in the analysis and solution of optimization problems. The idea is typically to derive an alternative formulation of a given problem, which is then easier to handle numerically or even analyt- ically. A particularly powerful framework is that of the so-called Wolfe Dual for quite general problems [Wol61], which is discussed in some detail in [Fle00] and [PAH19], for instance. The Wolfe Dual simplifies significantly in the case of so-called convex prob- lems which are of the form (A.26) with the additional requirements that a) the set IK = {x |gi(x) ≥0 for i = 1, 2, . . . n} is convex b) the objective function f(x) is a convex function on IK. In particular, condition (a) is satisfied if all functions gi(x) are convex or even linear. One of the most important results for convex optimization problem states that every local solution of the problem is also a global solution [Fle00,PAH19]. Under rather mild assumptions2 one can show that if x∗is a solution of (A.26), then x∗,λ∗ with the vector notation λ = {λ1, λ2, . . . λn} solves the following problem: Wolfe Dual of a convex problem of the form (A.26): maximize x∈Rd,λ∈Rn L x,λ subject to ∇xL(x,λ) = 0 and λ ≥0 (A.35) with the Lagrangian L as defined in Eq. (A.19). Moreover, the solution satisfies f(x∗) = L(x∗,λ∗). Note that while in the original problem (A.26) f is minimized with respect to x, here the maximization refers to λ and x under the constraint that ∇xL = 0. For a more detailed discussion of this subtlety and the Wolfe Dual of more general problems, see [Fle00,PAH19]. Example: quadratic optimization under linear constraints Frequently, the condition ∇xL(x,λ) = 0 can be used to eliminate the original variables x, resulting in a simplified optimization problem. As an example consider a quadratic problem with linear inequality constraints, which involves the vector of variables x ∈Rn, i.e. d = n, a symmetric matrix C ∈Rn×n and a vector of constants b ∈Rn: 2Most importantly, the functions f and gi should be continuously differentiable. An addi- tional so-called regularity assumption is discussed in [Fle00], sections 9.4 and 9.5. | 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, PAH19]. Here, the corresponding Wolfe Dual (A.35) reads Example: Wolfe Dual of the quadratic problem (A.36) maximize x∈Rn,λ ∈Rn 1 2x⊤C x −λ⊤(Cx −b) (A.37) subject to Cx = Cλ and λ ≥0. Now we can conveniently exploit the constraint Cx = Cλ in order to eliminate x. While it does not necessarily imply x = λ, we can still simplify the problem and obtain Example: Simplification of the Wolfe Dual (A.37) maximize λ ∈Rn −1 2 λ⊤C λ + λ⊤b subject to λ ≥0. (A.38) Note that this specific example has the exact same mathematical structure as the problem of optimal stability discussed in Section 3.7, if we set n = P and b = 1 ∈RP and moreover assume that C ∈RP ×P is defined according to Eq. (3.74). Renaming λ = x we recover the Wolfe Dual (3.97) for the problem of optimal stability, see Sec. 3.7. A.4 Gradient based optimization The gradient is the basis or at least an important component of many practical optimization techniques. Gradient descent persists to be one of the most popular and most successful method for the training of neural networks and more general machine learning setups. We discuss the reasons for this perhaps somewhat surprising fact in Chapter 5. A.4.1 Gradient and directional derivative The gradient as defined in Eq. (A.4) is closely related to so-called directional derivative. Consider a normalized vector a ∈Rd with |a| = 1. According to the | 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 = δik. In any direction with a⊤∇f(xo) < 0 the function decreases, while a with a⊤∇f(xo > 0 marks directions of ascent. In a given point xo the magnitude of the directional derivative quantifies how rapidly the function increases or decreases in the direction of a. For a given ∇f(xo) we see immediately that we obtain the directions of ... ... steepest ascent for a ∝+∇f(xo) ... steepest descent for a ∝−∇f(xo) ... stationarity for a ⊥∇f(xo). The gradient is always perpendicular to the level directions along which f is locally constant. A.4.2 Gradient descent The properties of the gradient can be used for the numerical minimization of a function f(x). Starting from an initial vector x(t = 0), we consider an iteration of the form x(t + 1) = x(t) −η ∇f(x(t)). (A.39) If the step size η > 0 is sufficiently small, the Taylor expansion of f in x(t) is dominated by the linear term and we have f(x(t + 1)) ≈f(x(t) −η|∇f(x(t))|2 < f(x(t)) (A.40) in every step of the sequence x(0) →x(1) . . . →x(t) →x(t + 1) . . . Consequently f can only decrease and the gradient descent (A.39) should ap- proach a local minimum. Analogously, we can devise gradient ascent with up- dates along +∇f(x(t)) in order to approach a local maximum. It is important that the step size η is small enough in order to guarantee validity of Eq. (A.40) and ensure convergence of the iteration. We will make this statement more precise in the following. The role of the step size We can investigate the behavior of gradient descent in greater detail by assuming that x(t) is already close to a local minimum x∗with f(x(t)) ≈f(x∗)+ 1 2 (x(t)−x∗)⊤H∗(x(t)−x∗) and ∇f(x(t)) ≈H∗(x(t)−x∗). (A.41) | 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 sides of Eq. (A.39) and approximated the gradient according to (A.41). Now we can exploit the fact that an orthonormal basis of Rd can be con- structed from the eigenvectors ui of H∗. Hence, we can expand the initial deviation as δ0 = d i=1 ci ui with coefficients ci ∈R in terms of the eigenvectors. We obtain immediately δt ≈[I −ηH∗]t δ0 = d i=1 ci [1 −ηρi]t ui The corresponding squared magnitude reads |δt|2 ≈ d i,j=1 cicj [1 −ηρi]t [1 −ηρj]t u⊤ i uj = d i=1 c2 i [1 −ηρi]2t where we have used that u⊤ i ui = 1 and u⊤ i uj = 0 for i ∕= j. We obtain that lim t→∞|δt|2 = 0 if and only if |1 −ηρi| < 1 for all i. Since all eigenvalues are positive and therefore 1−ηρi < 1 for all i, the (t →∞) asymptotic behavior of δt is dominated by the largest eigenvalue of the Hessian ρmax = max {ρ1, ρ2 . . . , ρN} : The condition for convergence of the iteration to the local minimum is that −1 < 1 −ηρmax ⇔η < ηmax = 2 ρmax . (A.42) For a finite range of step sizes 0 < η < ηmax, convergence is guaranteed, once the iteration is sufficiently close to the minimum. Moreover, we note that the factor (1 −ηρmax) changes sign for η = 1/ρmax. Hence, for η < ηmax/2 the iteration will approach the minimum smoothly, while for ηmax/2 < η < ηmax the orientation of δt flips in every gradient step and x(t) displays an oscillatory behavior while still approaching the minimum as t →∞. Finally, for η > ηmax, the iteration diverges and moves away from x∗in at least one eigendirection of H∗. The three different regimes are illustrated and summarized in Figure 5.6 of Chapter 5. It is important to realize that the above result is valid only in the vicinity of a given, local optimum. It does not enable us to make statements concerning the behavior or the role of the step size far away from an optimum. | 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. To a large extent, the initialization x(0) of a gradient procedure will determine which of the (potentially many) local minima will be approached. Close to a specific minimum, the local Hessian H(x(t)) can be seen as an estimate of H∗and used to select a suitable step size. More sophisticated higher order optimization techniques like Newton- or Quasi-Newton methods use H(x(t)) or an approximation thereof explicitly in the update, see for instance [Fle00,PAH19]. We refrain from introducing these more involved techniques here and restrict the discussion to relatively simple gradient-based methods which are frequently used in the context of machine learning. A.4.3 The gradient under coordinate transformations A frequently ignored property of the gradient is that the direction of steepest descent or ascent behaves non-trivially under coordinate transformations. In the context of the Adaline algorithm we see in Sec. 3.7.2 that gradient descent for the cost function of Eq. (3.77) in the space of embedding strengths x ∈RP is not equivalent to gradient descent in terms of the weights w ∈RN.3 Here we consider even simpler linear transformations from x ∈Rd to z ∈Rd : z = Ax with components zi = d j=1 Aijxj and A ∈Rd×d. In the following we use the notation ∇x = (∂/∂x1, ∂/∂x2, . . . ∂/∂xd)⊤and ∇z analogously to indicate which set of variables the gradient refers to. Interpreting the objective as a function of z we obtain with the chain rule ∂f ∂xk = d i=1 ∂f ∂zi ∂zi ∂xk = d i=1 Aik ∂f ∂zi i.e. ∇x f = A⊤∇z f, (A.43) or, assuming that A is invertible: ∇z f = A−⊤∇x f, while z = Ax. The transformation of the gradient is different from that of “ordinary” vec- tors, see [Tou12] for a brief discussion. The direction of steepest descent in the transformed coordinates does not coincide with the naive projection of the original gradient, in general. Therefore, we should be aware that a gradient descent procedure found for one coordinate system does not necessarily follow the steepest descent in an- other. This seems to cast some doubt on the distinguished role of gradient descent. We note however, reassuringly, that directions of descent need not be the steepest in order to be useful in numerical minimization procedures. 3It can be formulated, however, as gradient descent w.r.t. a different cost function in w. | 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 pioneered by Shun-ichi Amari in the context of machine learning, see [Ama98] for a discussion and further references. A.5 Variants of gradient descent The previous section presents and discusses plain “classical” gradient descent for the minimization of cost functions in unrestricted optimization problems. The basic idea of gradient descent can of course be modified and adapted to the specific properties and needs of a given problem. In the following we briefly discuss a number of variants which are particularly relevant in machine learning problems. This is the case, for instance, for the popular Stochastic Gradient Descent and its modifications and extensions. A.5.1 Coordinate descent Quite generally we consider the minimization of a continuously differentiable function f(x) of d-dimensional arguments x = (x1, x2, . . . xd). As discussed, the negative gradient −∇xf|x=x(t) marks the direction of the steepest descent in a given point x(t). We use the shorthand (. . .)|t for (. . .)|x=x(t) in the following. From a practical point of view it is often advantageous to resort to some direction of descent a(t) which is not necessarily the steepest and, therefore, only has to satisfy a(t)⊤∇f|t < 0. (A.44) A corresponding iterative procedure of the form x(t+1) = x(t) + η a(t) can be used to approach a (local) minimum of f, provided a suitable step size η is chosen. A particularly simple example is the sequential performance of steps along one of the coordinate axes with a(t) ∝ei(t). It is straightforward to see that we can easily satisfy Eq. (A.44) by setting a(t) = − e⊤ i(t) ∇f|t ei(t) = − ∂f ∂xi(t) t ei(t) = 0, 0, . . . −∂f ∂xi(t) t , 0, . . . 0 . Hence, the iteration step changes x only in one component, i(t), and it is deter- mined by the corresponding partial derivative of f. In deterministic-sequential coordinate descent a recursion of the form i(0) = 1; i(t) = [i(t−1)mod d] + 1 for t = 1, 2, 3, . . . generates a cyclic sequence of the form i(t) = 1, 2, 3, . . . , d, 1, 2, 3, . . . , d, 1, 2, . . . | 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 previous loop are used. The resulting, so-called coordinate-wise descent or coordinate descent for short, is a simple, sometimes surprisingly efficient method for minimization4. A recent review of this classical approach can be found in, e.g., [Wri15]. Obviously, one could also consider modifications with randomized selection of the updated coordinate, etc. A.5.2 Constrained problems and projected gradients In general, the solution of constrained problems by means of gradient descent techniques requires considerable refinement. One important approach in this context is gradient projection [Fle00]: As- sume that the search space is restricted to a region D by equality and/or in- equality constraints. If a step according to naive, unconstrained gradient descent yields x(t+1) = x(t)−η∇f ∈D, the step is accepted and x(t+1) = x(t). Other- wise one determines x(t+1) as a projection into D as x(t+1) = argminD ||x−x|| in an auxiliary optimization step. Depending on the precise nature of the con- straints, the computation of the projected gradient can be costly. In the case of simpler constraints, for instance as given by linear inequali- ties, one can often resort to so-called active set methods [Fle00,PAH19]. There, the iteration proceeds by unconstrained gradient descent or ascent until one or several constraints would be violated and, therefore, become active. Temporar- ily, active inequalities are treated as equality constraints in the following steps which move along the corresponding planes. A.5.3 Stochastic gradient descent A specific modification of gradient descent may be applied when the cost func- tion can be written as a sum over a number of individual functions as in f(x) = M m=1 hm(x) with ∇xf(x) = M m=1 ∇xhm(x), (A.45) where the second identify follows from the linearity of the gradient operator. In machine learning, very often the training process is guided by a cost function which can be written as a sum over a given set of example data. In supervised learning, e.g. regression, it typically corresponds to an error measure evaluated with respect to the individual examples in the training set D = ξµ ∈RN, yµ ∈R P µ=1 . 4Of course, coordinate ascent can be formulated for maximization analogously. | 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 trained system, e.g. all adaptive weights and thresholds of a neural network. Obviously this is of the form (A.45) with variables W and individual functions eµ(W). The discussion in the following refers to this machine learning setup and notation, but carries over to more general problems of the type (A.45), Of course, the numerical minimization of E could be aimed at by applying standard gradient descent as for any other, more general objective function. In analogy to Eq. (A.39) the basic form of updates would be W(t + 1) = W(t) −η ∇W E(W(t)). (A.47) We note, however, that costs of the form (A.46) can be interpreted as an empir- ical average (. . .) = µ(. . .)/P over the data set, which corresponds to drawing examples from D with equal probability 1/P : E(w) = eµ(W). Accordingly, the gradient of E w.r.t. W can be written as a mean of individual gradients: ∇wE(W) = 1 P P µ=1 ∇W eµ(W) = ∇W eµ(W). This suggests to approximate the full gradient of E by computing a restricted empirical mean over a random subset of examples from D. As an extreme case, we consider one randomly selected, single example in an individual training step: µ(t) ∈ {1, 2, . . . , P} (randomly selected with equal probability 1/P) W(t + 1) = W(t) −η ∇W eµ(t)(W(t)) (A.48) where we denote the learning rate by η to potentially distinguish it from η in batch gradient descent. Obviously, a single update step is, in general, computa- tionally cheaper than the full gradient descent (A.47) for which a sum over all examples has to be performed. Individual update steps follow a rough, stochastic approximation of the true gradient of E, hence the term stochastic gradient descent (SGD) has been coined for the iterative procedure. In practice, we could actually draw (with replace- ment) a random example from D independently in each step. Frequently, up- dates are organized in epochs, e.g. by generating a random permutation of {1, 2, . . . P} and presenting the entire D in this order before moving on to the next epoch with a novel randomized order of the examples. | 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 temporarily, as individual terms ∇W eµ can point uphill w.r.t. E with (∇W eµ) · ∇W E > 0. Let us now study the behavior of SGD near or in a local minimum W ∗with ∇W E(W ∗) = 0. Assuming that W(t) = W ∗we have that W(t + 1) = W(t) + ∆W(t) with ∆W = −η ∇W eµ(W ∗) for a randomly selected µ ∈{1, 2, . . . , P}. It follows that on average over the selection process 〈∆W(t)〉D = −η E(W ∗) = 0. In this sense, the SGD training becomes stationary in a local minimum of E and 〈∆W〉D →0 as w →W ∗. However, it is important to realize that, generally, in an individual update step, the weight vector will change even if W = W ∗is exactly satisfied. This can be seen from the average magnitude of the update step (∆W)2 D = η2 eµ(W ∗) 2 D = η2 1 P P µ=1 ∇W eµ(W ∗2 ≥0. Note that 〈(∆W)2〉D = 0 is possible iffall individual terms ∇W eµ(W ∗) = 0, which could mean that each contribution eµ is minimized in W ∗individually. If, for instance, a quadratic error measure of the form eµ = (σµ −τ µ)2/2 with ∇W eµ = (σµ −τ µ) is employed, this would imply that σµ = τ µ for all examples simultaneously, representing a perfect solution with E(W ∗) = 0. In general, however, 〈(∆W)2〉D > 0 in the local minimum W ∗. This indicates that for constant learning rate η > 0, the adaptive quantities W(t) will persist to fluctuate in the vicinity of a local minimum, even in the limit t →∞. The generic behavior for η > 0 is illustrated in Fig. 5.7 of Chapter 5. As discussed there, Robbins and Monro [RM51], see also [Bis95a, HTF01], have shown that convergence can be achieved with a time dependent learning rate schedule with lim t→∞η(t) = 0 which satisfies the conditions (5.16): (I) lim T →∞ T t=0 η(t)2 < ∞ and (II) lim T →∞ T t=0 η(t) →∞. (A.49) Intuitively, the first condition (I) states that η(t) has to decrease fast enough in order to achieve a stationary configuration, eventually. Condition (II) implies that the decrease is slow enough so that the entire search space can be explored efficiently without stopping the iteration too early. Simple schedules which reduce the learning rate asymptotically like η(t) ∝ 1/t for large t satisfy both conditions in (5.16). Just one possible and popular realization of such a decrease is of the form η(t) = a b + t with parameters a, b > 0. | 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 steps, for instance. Learning rate adaptation is frequently combined with so-called momentum terms which comprise infor- mation about previous updates. Up-to-date textbooks such as [GBC16] provide more details and further references. For a brief discussion, see Chapter 5, where also other modifications of SGD are introduced. A.6 Example calculation of a gradient In order to exemplify the computation of gradients in a feed-forward network, we consider here a specific two-layer architecture with N-dimensional input, K hidden units and a single output σ(ξ) = h K j=1 vj g w(j) · ξ . (A.50) All input-to-hidden weight vectors w(j) and hidden-to-output weights vj are assumed to be adaptive, while for simplicity local thresholds are not considered here. The output activation h(. . .) is taken to be (potentially) different from the hidden unit activations g(. . .). For a given data set D = {ξµ, τ µ}P µ=1, we consider the familiar quadratic deviation E = 1 P P µ=1 eµ with single example terms eµ = 1 2 σ(ξµ) −τ µ2 (A.51) Note that only σ depends on the weights, the target values τ are fixed and given in the data set. In the following we work out derivatives for one example term eµ only. For convenience, we omit the index µ and write σ in short for σ(ξ). First we compute the derivative with respect to one of the hidden-to-output weights. Only one term in the sum K j=1 . . . depends on vk and we obtain ∂e ∂vk = (σ −τ) ∂σ ∂vk = (σ −τ) h′ K j=1 vjg(w(j) · ξ) shorthand: δ g(w(k) · ξ). (A.52) Of course, h′ has to be specified in a concrete setting. For instance, if h(x) = tanh(γx) we have h′(x) = γ 1 −tanh2(γx) . Next, we take the derivative with respect to a single input-to-hidden weight w(m) n , i.e. the n-th component of the m-th input-to-hidden weight vector. With the same shorthand δ as defined above we obtain ∂e ∂w(m) n = (σ −τ) ∂σ ∂w(m) n = δ vm g′ w(m) · ξ ξn. (A.53) | 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 correspond to derivatives w.r.t. single weights. We could build a single gradient vector from the component-wise results by combining all adaptive quantities into a vector W as introduced in the previous section. It appears more natural, however, to write for m = 1, 2, . . . K: ∇w(m) E = δ vm g′ w(m) · ξ ξ (A.54) where the notation ∇w(m) stands for the gradient with respect to the m-th input-to-hidden weight vector. Note that both sides of the equation obviously correspond to N-dim. vectors. The partial derivatives w.r.t. the vk are given by the K additional equations (A.52). For the gradient of the full cost function we have to perform sums over all examples. Note that the abbreviation δ is defined for a particular single example, as well. Hence we get ∂E ∂vk = 1 P P µ=1 σ(ξµ −τ µ)h′ K j=1 vjg(w(j) · ξµ) shorthand: δµ g w(k) · ξµ (A.55) ∇w(m)E = 1 P P µ=1 δµ vm g′ w(m) · ξµ ξµ. (A.56) In a network with more layers, the chain rule has to be applied several times and in each layer terms similar to δ from previous layers appear. While the network response σ is determined by propagating an input towards the output, the gra- dient is computed by propagating the deviation (σ −τ) backwards through the network. In both operations the same weights play the role of coefficients. This is the basic concept behind the famous Backpropagation of Error for efficient gradient calculation in multi-layered networks [RHW86]. | 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 1.5 Feed-forward neural networks . . . . . . . . . . . . . . . . . . . . 14 2.1 Simple linear regression (Hubble diagram) . . . . . . . . . . . . . 25 3.1 The Mark I Perceptron . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Single layer perceptron . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Geometrical interpretation of the perceptron . . . . . . . . . . . 35 3.4 Rosenblatt perceptron algorithm . . . . . . . . . . . . . . . . . . 40 3.5 Linear separability in one dimension . . . . . . . . . . . . . . . . 47 3.6 The number of lin. sep. functions for N = 2 . . . . . . . . . . . . 48 3.7 Counting lin. sep. dichotomies (I) . . . . . . . . . . . . . . . . . 49 3.8 Counting lin. sep. dichotomies (II) . . . . . . . . . . . . . . . . . 50 3.9 Counting lin. sep. dichotomies (III) . . . . . . . . . . . . . . . . . 51 3.10 The fraction of lin. sep. functions . . . . . . . . . . . . . . . . . . 52 3.11 The pizza connection . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.12 Perceptron student-teacher scenario . . . . . . . . . . . . . . . . 56 3.13 Dual geometrical interpretation of the perceptron . . . . . . . . . 57 3.14 Perceptron learning in version space . . . . . . . . . . . . . . . . 58 3.15 Generalization error as a function of α = P/N . . . . . . . . . . . 60 3.16 Version space for P < N . . . . . . . . . . . . . . . . . . . . . . . 61 3.17 Stability of the perceptron . . . . . . . . . . . . . . . . . . . . . . 64 3.18 Support vectors (linearly separable data) . . . . . . . . . . . . . . 79 4.1 Support vectors (soft margin) . . . . . . . . . . . . . . . . . . . . 89 4.2 Architecture of “machines” . . . . . . . . . . . . . . . . . . . . . . 90 4.3 Committee and parity machine . . . . . . . . . . . . . . . . . . . 92 4.4 Storage capacity of machines . . . . . . . . | 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 . . . . . . . . . . . . . . . . . . . . . . . 113 5.6 Gradient descent near a minimum . . . . . . . . . . . . . . . . . 118 5.7 Stochastic gradient descent near a minimum . . . . . . . . . . . . 121 5.8 Sigmoidal and related activation functions . . . . . . . . . . . . . 128 5.9 Unbounded and one-sided activation functions . . . . . . . . . . 129 5.10 Extreme Learning Machine . . . . . . . . . . . . . . . . . . . . . 133 5.11 Shallow auto-encoder . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.12 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.13 Pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.14 Neocognitron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.15 LeNet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.1 Nearest Neighbor and Nearest Prototype Classifiers . . . . . . . . 142 6.2 GMLVQ system and data visualization . . . . . . . . . . . . . . . 152 7.1 Bias–variance dilemma . . . . . . . . . . . . . . . . . . . . . . . . 156 7.2 Bias and variance, underfitting and overfitting . . . . . . . . . . . 159 7.3 The double descent phenomenon . . . . . . . . . . . . . . . . . . 162 7.4 Early stopping and weight decay . . . . . . . . . . . . . . . . . . 164 7.5 Dropout regularization . . . . . . . . . . . . . . . . . . . . . . . . 170 7.6 Representative training data . . . . . . . . . . . . . . . . . . . . . 172 7.7 Receiver Operating Characteristics (ROC) . . . . . . . . . . . . . 179 7.8 Confusion matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.1 Log-transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8.2 Big dipper and enormous kitchen . . . . . . . . . . . . . . . . . . 191 8.3 Low-dimensional manifold . . . . . . . . . . . . . . . . . . . . . . 193 8.4 Multi-dimensional scaling . | 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 214 A.1 Multi-dimensional extrema and saddle points . . . . . . . . . . . 226 | 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 likelihood, 213 Generalized Learning Vector Quantization (GLVQ), 147 Generalized Matrix Learning Vector Quantization (GMLVQ), 151 Generic iterative perceptron updates (embedding strenghts), 38 Generic iterative perceptron updates (weights), 37 Grandmother Neuron, 93 K-means algorithm, 207 Kernel AdaTron (sequential updates), 101 Learning Vector Quantization (LVQ1), 145 Lloyd’s algorithm, 207 MinOver algorithm, 65 Oja’s subspace algorithm, 202 Optimal Brain Damage (OBD), 168 Optimal Brain Surgery (OBS), 168 Pocket algorithm, 86 Rosenblatt perceptron, 39 Sanger’s rule, 202 Stochastic Gradient Descent, 119 Synthetic Minority Oversampling Technique (SMOTE), 219 Tiling-like learning in the parity machine, 92 247 | 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 extreme learning machine EM expectation-maximization (algorithm) FP, FN false positives, false negatives (counts) fpr, fnr false positive rate, false negative rate GLVQ generalized learning vector quantization GMLVQ generalized matrix relevance learning vector quantization GMM gaussian mixture model hom. homogeneous(ly) inh. inhomogeneous(ly) ICA independent component analysis IQR interquartile range KL Kullback-Leibler (divergence) KT Kuhn-Tucker (e.g. KT conditions, KT theorem, KT point) l.h.s. left hand side lin. sep. linearly separable LMS least mean squares LVQ learning vector quantization MAE mean absolute error MAR missing at random MAP maximum a posteriori (probability) MCAR mis ing comp etely at r ndom MNAR missing not at random MDS multi-dimensional scaling MICE multiple imputation by chained equations MSE mean squared error NPC nearest prototype classifier {classification} OBD optimal brain damage OBS optimal brain surgeon ODE ordinary differential equation PM parity machine 248 | 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 quantization SENS sensitivity SPEC specificity SGD stochastic gradient descent SMOTE synthetic minority oversampling technique SNE stochastic neighborhood embedding SOM self-organizing map SSE sum of squared errors SVM support vector machine t-SNE t-distributed stochastic neighborhood embedding TP, TN true positives, true negatives (counts) tpr, tnr true positive rate, true negative rate UMAP uniform manifold approximation and projection VC Vapnik-Chervonenkis, e.g. in VC-dimension VQ vector quantization w.r.t. with respect to WTA winner-takes-all 249 | 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 possible, links to accessible preprint versions are provided as an alternative. All of the provided online sources have been accessed in April 2023. However, the author cannot guarantee the correctness of the links and is not liable for potential copyright infringements on the corresponding websites. 250 | The+Shallow+and+the+Deep_Page_264_Chunk4165 |
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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 machine learning techniques, with a bias towards classification and regression. Other learning paradigms and many recent developments in, for instance, Deep Learning are not addressed or only briefly touched upon. Biehl argues that having a solid knowledge of the foundations of the field is essential, especially for anyone who wants to explore the world of machine learning with an ambition that goes beyond the application of some software package to some data set. Therefore, The Shallow and the Deep places emphasis on fundamental concepts and theoretical background. This also involves delving into the history and pre-history of neural networks, where the foundations for most of the recent developments were laid. These notes aim to demystify machine learning and neural networks without losing the appreciation for their impressive power and versatility. Michael Biehl is Associate Professor of Computer Science at the Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence of the University of Groningen, where he joined the Intelligent Systems group in 2003. He also holds an honorary Professorship of Machine Learning at the Center for Systems Modelling and Quantitative Biomedicine of the University of Birmingham, UK. His research focuses on the modelling and theoretical understanding of neural networks and machine learning in general. The development of efficient training algorithms for interpretable, transparent systems is a topic of particular interest. A variety of interdisciplinary collaborations concern practical applications of machine learning in the biomedical domain, in astronomy and other areas. | 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 at: https://scholarworks.uark.edu/oer Part of the Atomic, Molecular and Optical Physics Commons, Elementary Particles and Fields and String Theory Commons, Engineering Physics Commons, and the Other Physics Commons Citation Citation Gea-Banacloche, J. (2019). University Physics I: Classical Mechanics. Open Educational Resources. Retrieved from https://scholarworks.uark.edu/oer/3 This Textbook is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Open Educational Resources by an authorized administrator of ScholarWorks@UARK. For more information, please contact scholar@uark.edu. | 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Position, displacement, velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Reference frame changes and relative motion . . . . . . . . . . . . . . . . . . . . . . 14 1.4 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5.1 Motion with (piecewise) constant velocity . . . . . . . . . . . . . . . . . . . . 21 1.5.2 Addition of velocities, relative motion . . . . . . . . . . . . . . . . . . . . . . 24 iii | University Physics I Classical Mechanics_Page_6_Chunk4200 |
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