Learning patterns with kernels and learning kernels from patterns

Introduction

Theory of kriging and Gaussian process regression

Note that there exists a Gaussian vector with arbitrary mean and valid covariance matrix with unique distribution. Note that the conditional expectation in equation (1.1.8) is an unbiased estimator for 𝑋𝑡 provided measurement XD =yog has the same mathematical form as the simple kriging estimator.

Mathematical applications and interplay of GPR

This link can be interpreted as the solution to a two-player optimal recovery game [101, Ch. These measurement functions can be structured in a hierarchy, leading to a hierarchy of wavelets.

GPR in the context of applications

Due to the mathematical equivalence to GPR, the likelihood of the corresponding GP can be used for kriging covariance kernel learning [53,105]. A notable special case is leave-one-out cross-validation (LOOCV), which takes 𝑘 = 𝑛, that is, the size of the training set.

Additive Gaussian processes and generalized additive models

In each term of the sum, the dependence of 𝑡 depends only on a single component9, and (1.4.4) can be written as. Such a regression is an example of a generalized additive model (GAM), which is defined to have the form.

Introduction to pattern learning problems

The third row shows CIFAR-10 images from the frog, truck, truck, deer, and automobile classes. The bottom row shows CIFAR-100 images from the cattle, dinosaur, apple, boy and aquarium fish classes.

Figure 1.1: Illustrations showing (1) 𝑎 (2) 5 examples of ∇ 𝑢 (3) 5 examples of 𝑢 (4) 1 example of 𝜂 = 𝑢 + 𝜁 .

Patterns and kernels

Note that 𝐴(𝑘) is the stiffness matrix of the elements (𝜓( .. 𝑖 )𝑖∈I(𝑘) form a bi-orthogonal system in the sense that [𝜙(𝑘). Repeat amicro-localKMD (presented in Section 4). .2) of the signal𝑣(𝑖) to obtain a very accurate estimate of the phase/amplitude𝜃𝑘, 𝑎𝑘of their corresponding mode 𝑣𝑘 for all 𝑘 ≤ 𝑖 (this iteration can reach near machine precision when the instantaneous frequencies are separated).

Denoising

Introduction to the denoising problem

33] showed that thresholding the coefficients of the damaged signal𝑇 𝑥+𝜁in the basis formed by the singular value decomposition (SVD) of𝑇 (which can be calculated in (𝑁3) complexity) approximated the minimax recovery to a fixed multiplicative constant. Let 𝑢 be an unknown element of 𝑉𝑀, given the noisy observation 𝜂 =𝑢+𝜁, find the most accurate approximation of 𝑢 in the energy norm k · k. This problem is addressed by expressing 𝜂 in the guess transform fitted to operator L and applying a truncation to the series.

It has been theoretically proven that this method produces a recovery within a constant of the minimum optimal recovery [153].

Summary of operator-adapted wavelets

Hierarchy of operator-matched pre-wavelets Let(𝜓( .. 𝑖 )𝑖∈I(𝑘) be the hierarchy of optimal restoration splines associated with(𝜙(. This result shows that the operator-matched pre-wavelets transform defined by 𝑣 † (𝑢) = 𝑢(𝑘) is an optimal recovery in the sense of Theorem 2.2.3.1-2. The games used in the subsequent developments will use pre-Haar wavelets (as defined below) as measurement functions𝜙( . 𝑘 ).

This result is used in the proof of the denoising result shown in the next section.

Denoising by truncating the gamblet transform

Further, Theorem 2.2.5.1 shows that the recovery accuracy,𝑢(𝑘), at rate L is bounded by 𝐿2rate of L𝑢. The following theorem states that𝜂(𝑙 . †) is an approximately minimum recovery of , 𝛿 and whose value can be made clear using the estimates of [101]). Both figures show that (1)𝑣(𝜂)and∇𝑣(𝜂) are exact approximations of 𝑢and∇𝑢 (2) the accuracy of these approximations increases with the regularity of 𝑓.

Similar to the 𝑑 = 1 plots, the 𝑑 = 2 plots show the accuracy of the recovery of 𝑢 and ∇𝑢 and the positive impact of the regularity of 𝑓 on that accuracy.

Figure 2.1: [153, Fig. 1], the plots of 𝑎 , 𝑓 , 𝑢 , 𝜂 , the near minimax recovery 𝑣 ( 𝜂 ) = 𝜂 ( 𝑙

Comparisons

The module outputs an estimate 𝑎(𝜏, 𝜃low,𝑒, 𝑓) of the amplitude 𝑎low(𝜏) of the mode 𝑣𝑖 and a correction 𝛿𝜃(𝜏, 𝜃low,𝑓,𝑒,determining,𝑒) 𝜏) + 𝛿𝜃(𝜏, 𝜃low,𝑒) , 𝑓) of the estimated state phase function𝜃low,𝑒. In the middle third of the range, 𝑣𝑖,𝑒and𝑣𝑖 are visually indistinguishable due to the small restoration errors. The purpose of the algorithm is to learn the kernel 𝐺𝑎 in the set of kernels {𝐺𝑏.

Application of empirical mode decomposition and Hilbert-Huang transform to seismic reflection data.

Table 2.2: Comparison of the performance of denoising algorithms for 𝑑 = 2.

The Mode Decomposition Problem

Hilbert-Huang transform

The Hilbert-Huang Transform (HHT) consists of both EMD and an application of the Hilbert transform. The algorithm will use 𝑖 as the index of the outer loop (steps 4 to 22), which each step calculates one IMF. Each 𝑣𝑖, 𝑗 can be interpreted as residuals of the signal initialized as the original signal𝑣1,1=𝑣in step 3.

It illustrates how derived IMFs can be converted into form𝑎(𝑡)cos(𝜃(𝑡)) and local properties of the oscillation can be estimated.

Figure 3.2: Upper and lower envelopes of residual signal 𝑣 𝑖, 𝑗 are 𝑢 and 𝑙 . The mean of the envelopes is 𝑚

Synchrosqueezing transform

These modes can be converted to the form 𝑎(𝑡)cos(𝜃(𝑡)) with the Hilbert transform, which will be presented in the next section. In applications, synchrosqueezing can be applied to signals not in this class, such as signals corrupted by noise, although theoretical accuracy limits would not apply. As can be observed in Figure 3.3.2, this transformation has a relatively large norm when the frequency scale, 𝑎, approximately aligns with the instantaneous frequency of a state, i.e. 𝜃0.

As can be seen in Figure 3.3.3, this leads to a more concentrated view of instantaneous frequencies.

Figure 3.3: Signal 𝑣 is the composition of 3 modes where the following is plotted in time-frequency domain: (1) the instantaneous frequencies of each mode (2) the norm of the continuous wavelet transform (CWT) of 𝑣 , | 𝑊

Extensions and further approaches

26], max-clamping with the baseline ECG waveform and deriving the instantaneous phase estimates 𝜃𝑖,𝑒. We will present a few variants of the KF algorithm, parametric and nonparametric, starting with the earlier one [103, Sec. The size of the convolution kernel is shown in the second and third columns from the left.

Specifically, given a random miniseries (𝑋𝑏, 𝑌𝑏) and a (random subsampling) half subseries (𝑋𝑐, 𝑌𝑐), we evolve 𝜃 and 𝛾6 in the direction of steepest loss descent.

Figure 4.1: [102, Fig. 23], (1) triangle base waveform (2) EKG base waveform.

Iterated micro-local kernel mode decomposition for known base

Max-pooling and the lowest instantaneous frequency

It will be applied in a module that identifies the phase and instantaneous frequency of the lowest frequency mode [102, Sec. Note that 𝜏 indicates the time from the center of the wave within [−1,1], while 𝜔 represents the frequency. The energy of the signal in (𝜏, 𝜔)-space is then defined by. 4.1.4) We define simulating the instantaneous phase and frequency estimation in SST.

Then let 𝐴low be defined as a subset of the time-frequency domain (𝜏, 𝜔) identified (as in Figure 4.4.2) as a narrow sausage band around the lowest instantaneous frequency defined by the local maxima of the S (𝜏, 𝜔, 𝑓). 4.1.7) be the estimated instantaneous frequency of the mode with the lowest instantaneous frequency and, with 𝜃𝑒defined as in (4.1.5), let.

The micro-local KMD module

4.2.7) Straightforward linear algebra together with (4.2.6) establishes that the vector𝑍(𝜏, 𝜃𝑒, 𝑓) can be computed as the solution of the linear system. See subfigures (1) and (2) at both the top and bottom of Figure 4.5 for illustrations of the windowed signal 𝑓𝜏(𝑡) and its projection slime𝜎↓0E. To use these formulations to construct the module, assume that the signal is a single state.

Ως εκ τούτου, θα χρησιμοποιήσουμε το 𝑎(𝜏, 𝜃𝑒, 𝑓) για να υπολογίσουμε το πλάτος𝑎(𝜏) της κατάστασης που αντιστοιχεί στην εκτίμηση 𝜃𝑒and𝛿𝜃(𝜏, 𝜃 , 𝑎 για την εκτίμηση της αλήθειας 𝜏 ) ≈ 𝜃𝑒(𝜏) +𝛿𝜃(𝜏, 𝜃𝑒, 𝑓).

Figure 4.5: [102, Fig. 28], top: 𝑣 is as in Figure 4.2 (the base waveform is trian- trian-gular)

The iterated micro-local KMD algorithm

Since 𝑐1 is known, this estimate produces the estimate𝑎low,𝑒𝑦¯(𝜃low,𝑒) for the overtones of the lowest mode. This residual is the sum of the estimate of the isolated base frequency component of 𝑣𝑗 and Í. See subfigures (3) and (5) of the top and bottom of Figure 4.5 for the results of peeling the first two estimated modes of the signal 𝑣 corresponding to both Figures 4.2 and 4.3 and subfigures (4) and (6) ) for the results of the corresponding projections in (4.2.5).

See subfigures (3) and (4) at the top and bottom of Figure 4.6 for the amplitude and its estimate of the results of peeling the first estimated mode and subfigures (5) and (6) corresponding to peeling the first two estimated modes of the signal 𝑣corresponding to both figures 4.2 and 4.3.

Numerical experiments

Let𝑇 ⊂ [−1,1] be the finite set of values ​​of 𝜏in the numerical discretization of the time axis with𝑁 :=|𝑇|elements. Conventional training methods involve optimizing a loss function depending on the final output of the ANN. The middle block shows layer specifications, and the shape of the output from each layer is on the right.

Deep regulation and direct training of inner layers of neural networks with kernel currents.

Iterated micro-local kernel mode decomposition for unknown base

Micro-local waveform KMD

The proposed approach is a direct extension of the approach presented in Section 4.2, and the shaded part of Figure 5.3 shows the new block that will be added to Algorithm3, the designed algorithm. As described below, this new block creates an estimator 𝑦𝑖,𝑒 of waveform 𝑦𝑖 from an estimate of 𝜃𝑖,𝑒 phase𝜃𝑖. When the assumed phase function 𝜃 :=𝜃𝑖,𝑒 is close to the phase function 𝜃𝑖 of the 𝑖 mode of the signal𝑣in the expansion (5.0.1), 𝑐𝑘 , 𝑗(𝜏, 𝜃𝑐𝑖,𝑒, 𝑣) gives the estimate of the Fourier coefficient 𝑖,(𝑘 , 𝑗) 5.0.2) 𝑖th basic waveform𝑦𝑖 at time𝑡 =𝜏.

Let 𝐼max be a maximizer of the function 𝐼 → 𝑁𝐼 over intervals of fixed width 𝐿, and define the estimate.

Figure 5.3: [102, Fig. 32], high level structure of Algorithm 4 for the case when the waveforms are unknown.

Iterated micro-local KMD with unknown waveforms algorithm

This quantity provides a good choice for 𝐿 and depends mainly on the choice of 𝛼and slightly on𝜔. As illustrated in Figure 5.3, we first identify the lowest frequency of the cosine component of each mode (steps 6 and 7 in Algorithm4). Then, from steps 10 to 18, we run a similar refinement loop as in Algorithm 3 with the addition of an application of micro-local KMD waveform in steps 15 and 16 to estimate the basic waveforms.

Finally, once each mode is identified, we re-apply the waveform estimation in steps 29-30 (after nearly removing the other modes and reducing overtone noise for better accuracy).

Numerical experiments

Further work in kernel mode decomposition

Recall that the parametric version of the KF algorithm uses a parameterized family of kernels𝑘𝜽 and optimizes the interpolation accuracy,𝜌, with respect to𝜽. 6.2.2) Figure 6.3 shows the KF4 flow of each of 𝑁 =250 Swiss Roll points at different stages of the algorithm. The ANN output, 𝑓𝜃(𝑥), lies in the domain R𝑛cl, where 𝑛cl is the number of classes in the classification problem.

Finally, we compare the KF loss, the LKF, and the ratio of between-class and within-class Euclidean distances on the output of the final convolution layers within each batch in Figure 7.3.

Kernel Flows

Parametric KF Algorithm

As input, in step 1, the parametric variant of the KF algorithm takes a training dataset(𝑡𝑖, 𝑦𝑖)𝑖, a parametric family of kernels𝑘𝜽 and an initial parameter𝜽0. We define𝜌(𝜽,Df,Yf,Dc,Yc) to be the squared relative error (in RKHS norm1 k · k𝑘𝜽 defined by𝑘𝜽) between the interpolants𝑢†, 𝑓 and𝑢† of the two nested and obtained subsets of the kernel𝑘𝜽, i.e. 2. 6.1.1) Note that tab𝜌 is doubly randomized through the selection of batches (Df, Yf) and sub-batches (Dc, Yc). Let us now consider the case where the interpolation points form a random subset of the discretization points.

The lack of smoothness of plots of 𝜌(𝑏), 𝑒(𝑏) vs 𝑛 stems from re-sampling the set 𝑍 at each step𝑛.

Figure 6.1: [103, Fig. 5], (1) 𝑎 (2) 𝑓 (3) 𝑢 (4) 𝜌 ( 𝑎 ) and 𝜌 ( 𝑏 ) (where 𝑏 ≡ 1) vs 𝑘 (5) 𝑒 ( 𝑎 ) and 𝑒 ( 𝑏 ) vs 𝑘 (6) 20 random realizations of 𝜌 ( 𝑎 ) and 𝜌 ( 𝑏 ) (7) 20 random realizations of 𝑒 ( 𝑎 ) and 𝑒 ( 𝑏 ) .

Non-parametric kernel flows

In our method, we construct the loss function to be the weighted sum of conventional and KF loss functions with kernel dependent on the outputs of the ANN inner layer. Thus, our loss function optimization is a novel technique for simultaneously training the outputs of multiple ANN layers. Additionally, DO [125] randomly removes components within each layer of the linear network map during training.

Using notations from the previous section, the outputs of the convolution layers, which include ReLU and pooling, are ℎ(1)(𝑥)toℎ(6)(𝑥)with output shapes described in the left column. The middle part shows block specifications, such as filter width and depth in each block, and the shape of the output from each layer is on the right. The performance of cross-validation and maximum likelihood estimators of spline smoothing parameters. Journal of the American Statistical Association.

Figure 6.3: [103, Fig. 10], 𝐹 𝑛 ( 𝑥 𝑖 ) for 8 different values of 𝑛 .

Kernel Flows Regularized Neural Networks