HighDimensional Separability for One and FewShot Learning
Abstract
:1. Introduction
1.1. AI Errors and Correctors
 The mistakes can be dangerous;
 Usually, it remains unclear who is responsible for them;
 The types of errors are numerous and often unpredictable;
 The real world is not a good i.i.d. (independent identically distributed) sample;
 We cannot rely on a statistical estimate of the probability of errors in real life.
 To preserve existing skills we must use the full set of training data;
 This approach requires significant computational resources for each error;
 However, new errors may appear in the course of retraining;
 The preservation of existing skills is not guaranteed;
 The probability of damage to skills is a priori unknown.
1.2. One and FewShot Learning
 If the feature space is effectively reduced, then the challenge of large dataset can be mitigated and we can rely on classical linear or nonlinear methods of statistical learning.
 If the data points in the latent space form dense clusters, then position of new data with respect to these clusters can be utilised for solving new tasks. We can also expect that new data may introduce new clusters, but persistence of the cluster structure seems to be important. The clusters themselves can be distributed in a multidimensional feature space. This is the novel and more general setting we are going to focus on below in Section 3.
1.3. Bibliographic Comments
 Blessing of dimensionality. In data analysis, the idea of blessing of dimensionality was formulated by Kainen [27]. Donoho considered the effects of the dimensionality blessing to be the main direction of the development of modern data science [28]. The mathematical backgrounds of blessing of dimensionality are in the measure concentration phenomena. The same phenomena form the background of statistical physics (Gibbs, Einstein, Khinchin—see the review [25]). Two modern books include most of the classical results and many new achievements of concentration of measure phenomena needed in data science [44,45] (but they do not include new stochastic separation theorems). Links between the blessing of dimensionality and the classical central limit theorems are recently discussed in [49].
 AI errors. The problem of AI errors is widely recognised. This is becoming the most important issue of serious concern when trying to use AI in real life. The Council of Europe Study report [10] demonstrates that the inevitability of errors of datadriven AI is now a big problem for society. Many discouraging examples of such errors are published [50,51], collected in reviews [52], and accumulated in a special database, Artificial Intelligence Incident Database (AIID) [53,54]. The research interest to this problem increases as an answer of the scientific community to the request of AI users. There are several fundamental origins of AI errors including uncertainty in training data, uncertainty in training process, and uncertainty of real world—reality can deviate significantly from the fitted model. The systematic manifestations of these deviations are known as concept drift or model degradation phenomena [55].
 AI correctors. The idea of elementary corrector together with statistical foundations was proposed in [30]. First stochastic separation theorems were proved for several simple data distributions (uniform distributions in a ball and product distributions with bounded support) [31]. The collection of results for many practically important classes of distributions, including convex combinations of logconcave distributions is presented in [13]. Kernel version of stochastic separation theorem was proved [36]. The stochastic separation theorems were used for development of correctors tested on various data and problems, from the straightforward correction of errors [32] to knowledge transfer between AI systems [56].
 Data compactness. This is an old and celebrated idea proposed by Braverman in early 1960s [57]. Several methods of measurement compactness of data clouds were invented [58]. The possibility to replace data points by compacta in training of neural networks was discussed [59]. Besides theoretical backgrounds of AI and data mining, data compactness was used for unsupervised outlier detection in high dimensions [60] and other practical needs.
1.4. The Structure of the Paper
 Correlation transformation that maps the dataspace into crosscorrelation space between data samples:
 PCA;
 Supervised PCA;
 Semisupervised PCA;
 Transfer Component Analysis (TCA);
 The novel expectationmaximization Domain Adaptation PCA (‘DAPCA’).
2. Postclassical Data
3. Stochastic Separation for FineGrained Distributions
3.1. Fisher Separability
 The probability distribution has no heavy tails;
 The sets of relatively small volume should not have large probability.
 The finite set Y in Theorem 1 is just a finite subset of the ball ${\mathbb{B}}_{n}$ without any assumption of its randomness. We only used the assumption about distribution of $\mathit{x}$.
 The distribution of $\mathit{x}$ may deviate significantly from the uniform distribution in the ball ${\mathbb{B}}_{n}$. Moreover, this deviation may grow with dimension n as a geometric progression:$$\rho (\mathit{x})/{\rho}_{\mathrm{uniform}}\le C/{r}^{n},$$
3.2. Granular Models of Clusters
 Volume of a ball with radius $1\varsigma $ is ${V}_{n}({\mathbb{B}}_{n}){(1\varsigma )}^{n}$. therefore for probability of $\mathit{x}$ belong to this ball, we have$$\mathbf{P}((\mathit{x},\mathit{x})\le 1\varsigma )\le {\rho}_{max}{V}_{n}({\mathbb{B}}_{n}){(1\varsigma )}^{n};$$
 For every $\mathit{z}\in {\mathcal{E}}_{0}$,$$\mathbf{P}((\mathit{x},\mathit{z})\ge \vartheta )\le {\rho}_{max}\frac{1}{2}{V}_{n}({\mathbb{B}}_{n}){(\sqrt{1{\vartheta}^{2}})}^{n};$$
 For every $\mathit{e}\in {\mathcal{E}}_{i}$$$\mathbf{P}\left((\mathit{x},\mathit{e})\ge \frac{\vartheta}{\sqrt{k}{d}_{i}}\right)\le {\rho}_{max}{V}_{n}({\mathbb{B}}_{n}){\left(\sqrt{1{\left(\frac{\vartheta}{\sqrt{k}{d}_{i}}\right)}^{2}}\right)}^{n}.$$
3.3. Superstatistic Presentation of ‘Granules’
3.4. The Superstatistic form of the Prototype Stochastic Separation Theorem
3.5. Compact Embedding of Patterns and Hierarchical Universe
4. MultiCorrectors of AI Systems
4.1. Structure of MultiCorrectors
 The correction system is organised as a set of elementary correctors, controlled by the dispatcher;
 Each elementary corrector ‘owns’ a certain class of errors and includes a binary classifier that separates situations with a high risk of these errors, which it owns, from other situations;
 For each elementary corrector, a modified rule is set for operating of the corrected AI system in a situation with a high risk of error diagnosed by the classifier of this corrector;
 The input to the corrector is a complete vector of signals, consisting of the input, internal, and output signals of the corrected Artificial Intelligence system, (as well as, if available, any other available attributes of the situation);
 The dispatcher distributes situations between elementary correctors;
 The decision rule, based on which the dispatcher distributes situations between elementary correctors, is formed as a result of cluster analysis of situations with diagnosed errors;
 Cluster analysis of situations with diagnosed errors is performed using an online algorithm;
 Each elementary corrector owns situations with errors from a single cluster;
 After receiving a signal about the detection of new errors, the dispatcher modifies the definition of clusters according to the selected online algorithm and accordingly modifies the decision rule, on the basis of which situations are distributed between elementary correctors;
 After receiving a signal about detection of new errors, the dispatcher chooses an elementary corrector, which must process the situation, and the classifier of this corrector learns according to a noniterative explicit rule.
 Simplicity of construction;
 Correction should not damage the existing skills of the system;
 Speed (fast noniterative learning);
 Correction of new errors without destroying previous corrections.
 Kernel versions of noniterative linear discriminants extend the area of application of the proposed systems, their separability properties were quantified and tested [36];
 Decision trees of mentioned elementary discriminants with bounded depth. These algorithms require small (bounded) number of iterations.
4.2. MultiCorrectors in Clustered Universe: A Case Study
4.2.1. Datasets
4.2.2. Tasks and Approach
 (Task 1) devise an algorithm to learn a new class without catastrophic forgetting and retraining, and;
 (Task 2) develop an algorithm to predict classification errors in the legacy classifier.
Algorithm 1: (Fewshot AI corrector [83]: 1NN version. Training). Input: sets $\mathcal{X}$, $\mathcal{Y}$; the number of clusters, k; threshold, $\theta $ (or thresholds ${\theta}_{1},\dots ,{\theta}_{k}$). 

Algorithm 2: (Fewshot AI corrector [83]: 1NN version. Deployment). Input: a data vector $\mathit{x}$, the set’s $\mathcal{X}$ centroid vector $\overline{\mathit{x}}$, matrices H, W, the number of clusters, k, cluster centroids ${\overline{\mathit{y}}}_{1},\dots ,{\overline{\mathit{y}}}_{k}$, threshold, $\theta $ (or thresholds ${\theta}_{1},\dots ,{\theta}_{k}$), discriminant vectors, ${\mathit{w}}_{i}$, $i=1,\dots ,k$. 

4.2.3. Results
4.2.4. Dimensionality and MultiCorrector Performance
5. Conclusions
6. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
AI  Artificial Intelligence 
i.i.d.  independent identically distributed 
ML  Machine Learning 
PCA  Principal Component Analysis 
TCA  Transfer Component Analysis 
DAPCA  Domain Adaptation PCA 
Appendix A. Elementary Preprocessing of Postclassical Data
Appendix A.1. Measure Examples by Examples and Reduce the Number of Attributes to dim(DataSet)
 Centralize data (subtract the mean);
 Delete coordinates with vanishing variance; (Caution: signals with small variance may be important, whereas signals with large variance may be irrelevant for the target task! This standard operation can help but can also impair the results).
 Standardise data (normalise to unit standard deviations in coordinates), or use another normalisation, if this is more appropriate; (Caution: transformation to the dimensionless variables is necessary but selection of the scale (standard deviation) affects the relative importance of the signals and can impair the results).
 Normalise the data vectors to unit length: ${\mathit{x}}_{i}\mapsto {\mathit{x}}_{i}/\parallel {\mathit{x}}_{i}\parallel $ (Caution: this simple normalisation is convenient but deletes one attribute, the length. If this attribute is expected to be important than it could be reasonable to use the mean value of $\parallel {\mathit{x}}_{i}\parallel $ that gives normalisation to the unit average length).
 Introduce coordinates in the subspace spanned by the dataset, $\mathrm{Span}\{{\mathit{x}}_{i}\}$ using projections on ${\mathit{x}}_{i}$.
 Each new data point $\mathit{y}$ will be represented by a Ndimensional vector of inner products with coordinates $(\mathit{y},{\mathit{x}}_{i})$.
Appendix A.2. Unsupervised, Supervised, and Semisupervised PCA
 Classical PCA, ${W}_{ij}\equiv 1$;
 Supervised PCA for classification tasks [104,105]. The dataset is split into several classes, ${K}_{v}$$(v=1,2,\dots ,r)$. Follow the strategy ‘attract similar and repulse dissimilar’. If ${\mathit{x}}_{i}$ and ${\mathit{x}}_{j}$ belong to the same class, then ${W}_{ij}=\alpha <0$ (attraction). If ${\mathit{x}}_{i}$ and ${\mathit{x}}_{j}$ belong to different classes, then ${W}_{ij}=1$ (repulsion). This preprocessing can substitute several layers of feature extraction deep learning network [93].
 Supervised PCA for any supervising task. The dataset for supervising tasks is augmented by labels (the desired outputs). There is proximity (or distance, if possible) between these desired outputs. The weight ${W}_{ij}$ is defined as a function of this proximity. The closer the desired outputs are, the smaller the weights should be. They can change sign (from classical repulsion, ${W}_{ij}>0$ to attraction, ${W}_{ij}<0$) or simply change the strength of repulsion.
 Semisupervised PCA was defined for a mixture of labelled and unlabelled data [106]. The data are labelled for classification task. For the labelled data, weights are defined as above for supervised PCA. Inside the set of unlabelled data the classical PCA repulsion is used.
Appendix A.3. DAPCA—Domain Adaptation PCA
 Select a family of classifiers in data space;
 Choose the best classifier from this family for separation the source domain samples from the target domain samples;
 The error of this classifier is an objective function for maximisation (large classification error means that the samples are indistinguishable by the selected family of classifiers).
 If ${\mathit{x}}_{i},{\mathit{x}}_{j}\in {K}_{v}$ then ${W}_{ij}=\alpha <0$ (the source samples from one class, attraction);
 If ${\mathit{x}}_{i}\in {K}_{u}$${\mathit{x}}_{j}\in {K}_{v}$ ($u\ne v$) then ${W}_{ij}=1$ (the source samples from different classes, repulsion);
 ${\mathit{x}}_{i},{\mathit{x}}_{j}\in \mathbf{Y}$ then ${W}_{ij}=\beta >0$ (the target samples, repulsion);
 For each target sample ${\mathit{x}}_{i}\in \mathbf{Y}$ find k closest source samples in $\mathbf{X}$. Denote this set ${E}_{i}$. For each ${\mathit{x}}_{j}\in {E}_{i}$, ${W}_{ij}=\gamma <0$ (the weight for connections of a target sample and the k closest source samples, attraction).
Appendix B. ‘Almost Always’ in InfiniteDimensional Spaces
 The sets of measure zero are negligible.
 The sets of Baire first category are negligible.
 A union of countable family of thin sets should be thin.
 Any subset of a thin set should be thin.
 The whole space is not thin.
 Almost always a function $f\in C(X)$ has nowhere dense set of zeros $\{x\in X\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}f(x)=0\}$ (the set of exclusions is completely thin in $C(X)$).
 Almost always a function $f\in C(X)$ has only one point of global maximum.
Appendix C. Flowchart of MultiCorrector Operation
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n  10  25  50  100  150  200 

$Y\le $  0.38  91  $8.28\times {10}^{5}$  $6.85\times {10}^{13}$  $5.68\times {10}^{21}$  $4.70\times {10}^{29}$ 
$\rho (\mathit{x})/{\rho}_{\mathrm{uniform}}\le $  2.86  13.9  194  $3.76\times {10}^{4}$  $7.30\times {10}^{6}$  $1.41\times {10}^{9}$ 
n  10  25  50  100  150  200 

$Y\le $  0.61  $9.5$  910  $8.28\times {10}^{6}$  $7.53\times {10}^{10}$  $6.85\times {10}^{14}$ 
n  25  50  100  150  200 

$Y\le $  $0.55$  30  $9.26\times {10}^{4}$  $2.81\times {10}^{8}$  $8.58\times {10}^{11}$ 
Layer Number  Type  Size 

1  Input  $32\times 32\times 3$ 
2  Conv2d  $4\times 4\times 64$ 
3  ReLU  
4  Batch normalization  
5  Dropout 0.25  
6  Conv2d  $2\times 2\times 64$ 
7  ReLU  
8  Batch normalization  
9  Dropout 0.25  
10  Conv2d  $3\times 3\times 32$ 
11  ReLU  
12  Batch normalization  
13  Dropout 0.25  
14  Conv2d  $3\times 3\times 32$ 
15  ReLU  
16  Batch normalization  
17  Maxpool  pool size $2\times 2$, stride $2\times 2$ 
18  Dropout 0.25  
19  Fully connected  128 
20  ReLU  
21  Dropout 0.25  
22  Fully connected  128 
23  ReLU  
24  Dropout 0.25  
25  Fully connected  9 
26  Softmax  9 
Attributes  ${\mathit{x}}_{1},\dots ,{\mathit{x}}_{9}$  ${\mathit{x}}_{10},\dots ,{\mathit{x}}_{137}$  ${\mathit{x}}_{138},\dots ,{\mathit{x}}_{265}$  ${\mathit{x}}_{266},\dots ,{\mathit{x}}_{393}$ 

Layers  26 (Softmax)  19 (Fully connected)  22 (Fully connected)  23 (ReLU) 
n  20  100  137  300  393 

$Y\le $  $1.21\times {10}^{2}$  $2.58\times {10}^{18}$  $9.21\times {10}^{25}$  $1.72\times {10}^{59}$  $1.65\times {10}^{78}$ 
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Gorban, A.N.; Grechuk, B.; Mirkes, E.M.; Stasenko, S.V.; Tyukin, I.Y. HighDimensional Separability for One and FewShot Learning. Entropy 2021, 23, 1090. https://doi.org/10.3390/e23081090
Gorban AN, Grechuk B, Mirkes EM, Stasenko SV, Tyukin IY. HighDimensional Separability for One and FewShot Learning. Entropy. 2021; 23(8):1090. https://doi.org/10.3390/e23081090
Chicago/Turabian StyleGorban, Alexander N., Bogdan Grechuk, Evgeny M. Mirkes, Sergey V. Stasenko, and Ivan Y. Tyukin. 2021. "HighDimensional Separability for One and FewShot Learning" Entropy 23, no. 8: 1090. https://doi.org/10.3390/e23081090