# Decomposition Methods for Machine Learning with Small, Incomplete or Noisy Datasets

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## Abstract

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## 1. Introduction

#### 1.1. ML with Low Quality Datasets: State-of-the-Art and Recent Progress

#### 1.1.1. Classical Data Augmentation

#### 1.1.2. Classical Approaches to ML with Incomplete Data

#### 1.2. Mathematical Notation and Definitions

## 2. Methods

#### 2.1. A Unified View of Data Decomposition Models for ML

#### 2.2. Subspace Approximation (PCA)

#### 2.3. Sparse Decomposition (SD)

#### 2.4. Empirical Mode Decomposition (EMD)

- Determine the local maxima and minima of the signal $x\left(t\right)$.
- Calculate the upper (lower) envelope by interpolating the local maxima (minima) points. The interpolation can be carried out in different ways (linear interpolation, spline interpolation, etc.), which could lead to slightly different results.
- Calculate the local mean $m\left(t\right)$ by averaging the upper and lower envelopes.
- Calculate the first IMF candidate ${h}_{1}\left(t\right)=x\left(t\right)-m\left(t\right)$.
- Checks whether candidate ${h}_{1}\left(t\right)$ meets the criteria to be an IMF:
- If ${h}_{1}\left(t\right)$ meets the criteria, define the first IMF as ${c}_{1}\left(t\right)={h}_{1}\left(t\right)$.
- If ${h}_{1}\left(t\right)$ does not meet the criteria, set $x\left(t\right)={h}_{1}\left(t\right)$ and repeat from step 1

#### 2.5. Tensor Decomposition (TD)

**Low-rank Tucker decomposition:**when the core tensor is much smaller than the original, i.e., ${R}_{n}\ll {I}_{n}$ [47,48] (see Figure 1e).

**Sparse Tucker decomposition:**when core tensor is of the same size or larger than tensor $\underline{\mathbf{X}}$ but it is sparse as illustrated in Figure 1g. In this case, by looking at Equation (7), we conclude that the Sparse Tucker model corresponds to the classical Sparse Coding model of (4) with a dictionary that is obtained as the Kronecker product of three mode dictionaries, i.e., $\mathsf{\Phi}={\mathbf{A}}_{3}\otimes {\mathbf{A}}_{2}\otimes {\mathbf{A}}_{1}$ [49,50]. Mode dictionaries can be chosen from classical sparsifying transforms such as wavelets, cosine transform and others or, if enough data is available, they can be learned from a dataset, which usually provides higher levels of sparsity and compression. A Kronecker dictionary learning algorithm was introduce in [50] and later a variant with orthogonality constraints was proposed in [51].

#### 2.6. Comparison of Methods for ML with Low-Quality Datasets

## 3. Results

#### 3.1. Brain Signal Classification

#### 3.1.1. BCI with Missing/Corrupted Measurements

#### 3.1.2. Efficient Data Augmentation for BCI

- Randomly select N frames from the set of frames belonging to the selected class.
- Decompose, using EMD, each one of the N frames, generating a set of IMFs per channel and frame.
- Then, select the first IMF from the first selected frame (one per channel and keeping the same position for each channel), the second IMF from the second selected frame, and successively until the Nth frame, which contributes with its Nth IMF.
- Add up all the IMFs corresponding to the same channel to build each new EEG channel of the new artificial frame.

#### 3.1.3. Epileptic Focal Detection with Limited Data

- Randomly choose seven iEEG signals from the dataset and apply the DCT to obtain the spectrum.
- Segment the spectrum into the seven physiological frequency bands (Delta: 0–4 Hz, Theta: 4–8 Hz, Alpha: 8–13 Hz, Beta: 13–30 Hz, Gamma: 30–80 Hz, Ripple: 80–150 Hz, and Fast Ripple: 150–Nyquist Hz), extract one frequency band of each of the decompositions, from lowest to highest frequencies, and merge the seven extracted components (frequency bands) to create a new artificial spectrum. For example, we can extract the delta, the theta, the alpha, the gamma, the ripple, and the fast ripple from the first, the second, the third, the fourth, the fifth, the sixth, and the seventh signal, respectively.
- Apply the inverse DCT to the artificial spectrum in the frequency domain to obtain an artificial signal in the time-domain.

#### 3.2. Classification of Noisy Faces

#### 3.3. Scada Data Completion in Water Networks

## 4. Conclusions and Discussion

- The decomposition methods reviewed in this work for imputation of missing/corrupted values do not exploit the class label information in a supervised learning scenario. A possible further improvement of current methods is to incorporate label information into the decomposition models. We believe that missing data values could be better recovered if the class label of the corresponding data sample is known.
- EMD based data augmentation was developed in an ad-hoc fashion. We believe that more theoretical insights could be explored allowing future improvements, for example, by re-designing the way that IMFs are calculated in order to produce class-preserving artificial samples.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Decomposition models. (

**a**)

**General linear model**: a collection of vector data samples organized as columns of a matrix dataset $\mathbf{X}$ is approximated by the product of matrices $\mathsf{\Phi}$ and $\mathbf{A}$. (

**b**)

**Subspace approximation**: all vectors in the dataset are approximated by linear combination of few vectors (principal components). (

**c**)

**Sparse coding**: each vector in the dataset is approximated by the linear combination of atoms (columns of a dictionary $\mathsf{\Phi}$). In both, (

**b**,

**c**), the optimal choice of matrix $\mathsf{\Phi}$ can be computed from the dataset itself by means of the SVD and an dictionary learning algorithm, respectively. (

**d**)

**EMD**: every single signal is decomposed as a sum of characteristic modes. Tensor decomposition models such as

**Low-rank Tucker**(

**e**),

**Low-rank CP**(

**f**) and

**Sparse Tucker**(

**g**) can be written as sum of rank-1 tensors (

**h**).

**Figure 2.**Training a BCI classifier (LDA) with noisy/missing EEG measurements. (

**a**) Preprocessing steps: first, the positions in which the data is missed/corrupted are identified; then, a mask is created to ignore values in those positions; and finally, the tensor model reconstructs the missing data. (

**b**) Results with randomly missing entries. (

**c**) Results with random missing channels. (Figure adapted from [57]).

**Figure 3.**EEG data augmentation: (

**a**) For each new EEG signal to be generated, N available EEG signals are randomly selected and their EMDs are computed. (

**b**) To generate an artificial EEG signal, IMFs from different signals are combined.

**Figure 4.**An artificial signal generation with the DCT. (

**a**) Seven intracranial iEEG signals at either focal or non-focal area. (

**b**) DCT coefficients in the spectrum-domain. The spectra are segmented into seven physiological sub-bands, and the sub-band components extracted from each spectrum are merged to create an artificial spectrum. (

**c**) The inverse DCT leads to the resulting artificial signal.

**Figure 5.**(

**a**) The new proposed approach to eliminate the noise and improve the classification accuracy is based on the GiT-BEMD decomposition. The high frequency IMFs are discarded and the (noiseless) image is reconstructed by summing up the rest of the modes. This is the image that will feed the classifier. (

**b**) Comparison of classification results using a Support Vector Machine (SVM) and K-Nearest Neighbor (kNN) classifiers applied to noisy, filtered faces (Gaussian, Mean, Median) and GiT-BEMD processed faces.

**Figure 6.**Data tensorization of a 3 week tensor with 200 samples of lost data bursts. In (

**a**) the green line shows the original data, and the red line shows the lost burst. The soft blue window shows the data introduced in burst-centered tensor, which forces the burst to be in the center of the window. Panels (

**b**) shows how the continuous flow of data in the soft blue window is fragmented to be allocated in the tensor as shown in panel (

**c**).

**Table 1.**Comparison of methods for Machine Learning (ML) problems with low-quality datasets. Article sections in which these methods are discussed are noted in the first column and relevant references are included in the last column.

Method | Characteristics | Shortcomings | Advantages | Application | References |
---|---|---|---|---|---|

Class preserving transforms (Section 1.1.1) | Ad-hoc; mostly images oriented but some extensions to other types of data were explored | Limited theory available; difficult to apply to arbitrary type datasets | Easy to use; widely available in deep learning platforms | Data augmentation | [7,8,9,10,11,12,13,14,15,16,17] |

Empirical Mode Decomposition (EMD) based data generation (Section 2.4, Section 3.1.2, Section 3.1.3 and Section 3.2) | Ad-hoc; based on the manipulation and recombination of Intrinsic Mode Functions (IMFs); | Lack of theoretical ground | Easy to use; capture dataset discriminative features; denoising power | Electroencephalography (EEG)/ invasive EEG (iEEG) data augmentation and denoising | [45,52,53,54,55] |

Transform domain based data generationSection 3.1.3) | Ad-hoc; based on the manipulation and recombination of spectrum domain components obtained by Discrete Cosine Transform (DCT), Wavelets, etc. | Lack of theoretical ground | Easy to use; capture dataset discriminative features | iEEG data augmentation | [45,52,53,56] |

Statistical imputation (Section 1.1.2) | Preprocessing step in ML; exploit statistical properties of datasamples; wide variety of methods, from simple ones (mean) to more sophisticated (regression, k-Nearest Neighbor (kNN), Self Organization Map (SOM), etc.) | Does not use the class label information of data samples | Computationally efficient | ML with incomplete or corrupted data | [18,19,20,21,22,23] |

Probabilistic modelling (Section 1.1.2) | Gaussian Mixture Model (GMM) as data model; Bayesian classification; fitting model and classifiers in an Expectation-Maximization (EM) fashion; can be adapted to deep neural networks | Computationally expensive | Incorporates class label information of data samples; elegant theoretical approach | ML with incomplete or corrupted data | [19,24,28] |

Low-rank matrix completion (Section 1.1.2) | Based on Singular Value Decomposition (SVD) | Computationally very expensive; not suitable for complex boundary functions | Incorporates class label information of data samples | ML with incomplete or corrupted data | [25,26,27] |

Tensor decomposition (TD) based imputation (Section 2.5, Section 3.1.1 and Section 3.3) | Preprocessing step in ML; based on low-rank TDs (e.g., Tucker, CANDECOMP/PARAFAC (CP), etc.) or sparse TDs | Does not use the label information of data samples | Exploits intricate relationship among modes in multidimensional data | ML with incomplete or corrupted data | [50,57,58,59,60,61,62] |

**Table 2.**Dispersion ratio R computed in seven subjects (S01-S07) with Equation (8) for right (R) and left (L) classes at different levels of used artificial frames (AF). Results with $R>3$ are highlighted in red and with $2<R<3$ in orange.

S01 | S02 | S03 | S04 | S05 | S06 | S07 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AF(%) | R | L | R | L | R | L | R | L | R | L | R | L | R | L |

2.5 | 0.12 | 0.67 | 0.22 | 0.64 | 0.58 | 1.27 | 0.32 | 0.31 | 0.32 | 0.27 | 0.33 | 0.64 | 0.34 | 0.69 |

5.0 | 0.05 | 1.03 | 0.82 | 0.56 | 1.11 | 1.02 | 0.46 | 0.45 | 0.18 | 0.35 | 0.47 | 0.83 | 0.01 | 0.63 |

7.5 | 0.29 | 0.88 | 1.03 | 0.07 | 1.06 | 1.51 | 0.51 | 0.51 | 0.00 | 0.02 | 1.17 | 1.49 | 0.46 | 0.62 |

10.0 | 0.37 | 1.13 | 0.99 | 0.11 | 1.19 | 1.75 | 0.80 | 0.46 | 0.38 | 0.08 | 1.04 | 1.66 | 0.49 | 0.84 |

12.5 | 0.24 | 0.94 | 1.42 | 0.04 | 1.89 | 1.86 | 1.00 | 0.44 | 0.46 | 0.27 | 0.87 | 1.52 | 0.40 | 0.85 |

25.0 | 0.09 | 1.44 | 2.79 | 0.44 | 2.13 | 1.94 | 1.28 | 0.61 | 0.96 | 0.78 | 0.71 | 2.09 | 0.51 | 1.28 |

37.5 | 0.11 | 1.55 | 3.12 | 0.41 | 1.97 | 2.01 | 1.20 | 0.69 | 1.07 | 1.18 | 0.57 | 2.66 | 0.73 | 1.92 |

50.0 | 0.15 | 1.45 | 2.86 | 1.00 | 2.18 | 2.68 | 1.27 | 1.06 | 1.42 | 1.23 | 0.62 | 2.76 | 0.73 | 1.86 |

**Table 3.**Algorithms’ performance in terms of the MSE per sample. Best results are indicated in bold text.

Method | Weeks | MSE/Sample | |
---|---|---|---|

Burst Length = 100 | Burst Length = 200 | ||

Forward & Backward Predictors | - | 1.11 | 2.23 |

SingleDecomp—CP | 3 | 0.87 | 1.78 |

SingleDecomp—CP | 7 | 0.80 | 1.58 |

SingleDecomp—TK | 3 | 0.80 | 1.43 |

SingleDecomp—TK | 7 | 0.71 | 1.28 |

DoubleDecomp—CP | 3 | 0.55 | 1.05 |

DoubleDecomp—CP | 7 | 0.52 | 1.02 |

DoubleDecomp—TK | 3 | 0.55 | 1.04 |

DoubleDecomp—TK | 7 | 0.50 | 0.97 |

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**MDPI and ACS Style**

Caiafa, C.F.; Solé-Casals, J.; Marti-Puig, P.; Zhe, S.; Tanaka, T.
Decomposition Methods for Machine Learning with Small, Incomplete or Noisy Datasets. *Appl. Sci.* **2020**, *10*, 8481.
https://doi.org/10.3390/app10238481

**AMA Style**

Caiafa CF, Solé-Casals J, Marti-Puig P, Zhe S, Tanaka T.
Decomposition Methods for Machine Learning with Small, Incomplete or Noisy Datasets. *Applied Sciences*. 2020; 10(23):8481.
https://doi.org/10.3390/app10238481

**Chicago/Turabian Style**

Caiafa, Cesar Federico, Jordi Solé-Casals, Pere Marti-Puig, Sun Zhe, and Toshihisa Tanaka.
2020. "Decomposition Methods for Machine Learning with Small, Incomplete or Noisy Datasets" *Applied Sciences* 10, no. 23: 8481.
https://doi.org/10.3390/app10238481