PMC

The fused feature of RMS² and SampEn based on (a) principal component analysis (PCA) and (b) Isometric Feature Mapping (Isomap).

The performance of nonlinear dimensionality reduction methods, locally linear embedding (LLE) and isometric feature mapping (Isomap), are shown in the box plot. Isomap marginally outperformed LLE with a median and standard deviation for dice similarity (DS) score, respectively, as 0.78 ± 0.09 and 0.74 ± 0.09.

Non-linear approaches to dimensionality reduction of prefiltered spectroscopic data: (a) isometric feature mapping (Isomap); (b) locally linear embedding (LLE); (c) spectral embedding (SE); (d) t-distributed stochastic neighbor embedding.

(a) Typical multiparametric MRI data and (b) resulting embedded images and scattergrams for a malignant breast case from three NLDR algorithms Diffusion Maps (DfM), Isometric feature mapping (ISOMAP), Locally Linear Embedding (LLE). The lower scattergram shown is derived from Isomap. Clear demarcation of the lesion and surrounding breast are shown.

Each row represents a classifier trained with a label excluded whose projection is visualised. While each column represents a different classifier architecture evaluated. A denser clustering represents a consistent view of the unknown class by the architecture.

(a) Typical axial multiparametric MRI data from a patient with no breast lesion. (b) resulting embedded images and scattergrams demonstrating the separation of fatty and glandular tissue in the embedded image, the scattergram shown is derived from Isometric feature mapping (ISOMAP).

(a) Unsorted point clouds. (b) Point clouds after sorting by order of phase in flagellar cycle, as identified by isometric feature mapping (Isomap). The first six rows of the sorted beat correspond to “recovery” as the primary bend propagates from base to tip; the last six rows roughly correspond to the “power stroke” driven by propagation of the reverse bend.

The plots for the MNIST dataset based on six dimensionality reduction approaches, including (A) Isomap, (B) LLE, (C) PCA, (D) MDS, (E) t-SNE, and (F) WDM-tSNE. MNIST, Modified National Institute of Standards and Technology; ISOMAP, Isometric Mapping; PCA, Principal Component Analysis; LLE, Locally Linear Embedding; MDS, Multidimensional Scaling; t-SNE, t-Distributed Stochastic Neighbor Embedding; WDM-tSNE, Weighted Distance Metric and the t-stochastic neighbor embedding algorithm.

Architecture diagram of the research framework. ANN artificial neural network, R² coefficient of determination, RMSE root mean square error, MAE mean absolute error, PCA principal component analysis, ISOMAP isometric mapping, HCA hierarchical cluster analysis.

The mechanism of Isomap.

Fuzzy c-means algorithm visualization on Isomap database.

3D visualization of ISOMAP based rockburst database.

(A) Multiparametric brain MRI data as input to nonlinear dimensionality reduction (NLDR) methods. (B-upper) Output embedded images from locally linear embedding (LLE) and isometric feature mapping (Isomap). (B-lower) The resulting binary masks are shown. (C-upper) The ground truth (contrast subtraction image). (C-lower) Corresponding binary image for ground truth, used to evaluate methods using dice similarity scores.

Confirmation of identified populations and their intercluster relationship. (A) Two-dimensional scatter graphs from a representative sample are shown for manual gating and confirmation of identified cell subpopulations of CD68⁻ and CD68⁺ clusters. The gating controls are based on the same population in T cells (red box) and are shown below the identified populations (black box). RM subsets denote the likely RMs, whereas TRM subsets denote the likely TRMs. (B) The intercluster relationship assessed by isometric feature mapping (Isomap) and principal component analysis (PCA) are shown.

Scatter plots of the dimensionality reduction algorithm projections applied on the baseline dataset. (A) The principal component analysis (PCA) scatter plot of the bleeding dataset and the cumulative percentage of variance as a function of the number of principal components (PCs), and (B) the resulting scatter plot for kernel PCA, isometric mapping (Isomap), and t-distributed stochastic neighboring embedding (t-SNE). The dotted line in panel (A) indicates that the >95% explained variance occurs after 40 PCs. Bleeders are plotted on top of the nonbleeders to make them visually discernable.

Different combinations of the methods in each section were adopted in different studies. Pre-processing and recognition methods used in commercial software were not included due to the lack of clarify on what algorithms were used in them. The face used in illustration was an average face generated from http://faceresearch.org/demos/average, which is available open access (CC-BY-4.0) []. The icons used in the figure are available open access (CC-BY) from the NounProject.com. 2D/3D: two/three-dimension, ANOVA: analysis of variance, AUROC: area under the receiver operating characteristic, CNN: convolutional neural network, IR: infrared, ISOMAP: isometric mapping, KNN: k nearest neighbor, LDA: linear discriminant analysis, ML: machine learning, RNN: recurrent neural network, ROI: region of interest, SVM: support vector machine.

Flowchart of this study. (a) preprocessing of dynamic susceptibility contrast perfusion-weighted imaging (DSC-PWI) datasets; (b) computing whole-brain dynamic radiomics features (DRF); (c) feature selection and combination strategy; (d) evaluating performance with ten learning models. The DRF in (b) are combined with the radiomics features of 3D images in the time series of DSC-PWI image; the five unsupervised feature selection are principal component analysis (PCA), independent component correlation algorithm (ICA), t-distributed stochastic neighbor embedding (TSNE), uniform manifold approximation and projection (UMAP), and isometric feature mapping (ISOMAP); the ten models are support vector machine (SVM), decision tree (DT), Adaboost classifier (Ada), neural network (NN), random forest (RF), k-nearest neighbors (KNN), logistic regression (LR), linear discriminant analysis (DA), gradient boosting classifier (GBDT), and GaussianNB (NB).

Summary of protein expression data. a In total, 5825 proteins were identified, with 990 quantified with only one or two spectra and which were thus omitted from our primary comparative analysis. The remaining 4835 proteins are classified as to whether they were quantified in six or fewer distinct regions. b Proportion of identified, quantified proteins showing a change in expression in Alzheimer’s disease (AD) in each of the six regions under study. c Heat map and dendrogram showing the relationship between protein expression in each region mapped using proteins present in all six regions, with three distinct ‘groups’ based on highly affected (hippocampus (HP), entorhinal cortex (ENT), cingulate gyrus (CG)), moderate (motor cortex (MCx), sensory cortex (SCx)) and spared (cerebellum (CB)) clearly visible. d Edwards–Venn diagram showing the overlap of protein expression changes between brain regions, including only proteins quantified in all regions. e Isometric mapping (Isomap) representation of protein expression data between brain regions showing correlation in protein expression from non-affected towards affected regions, with the exception of cerebellum, which shows distinct patterns of protein expression in AD

2D microbiomeprints of Disease-Set1 generated by random and manifold embedding algorithms and disease-prediction performances of the corresponding AggMapNet models
(A) The example of the different embedding algorithms for generating the 2D signatures in the IBD dataset. The random uniform embedding (RUE) and five manifold embedding approaches, including multi-dimensional scaling (MDS), isometric mapping (ISOMAP), locally linear embedding (LLE), t-distributed stochastic neighbor embedding (t-SNE), and Uniform Manifold Approximation and Projection (UMAP), are used for the embedding of the microbes to generate the microbial grid map and 2D microbiomeprint Fmaps. The metagenomic-based hierarchical clustering is used to group the microbes into ten subgroups. Each color of microbes is one cluster from the clustering of the microbial species.
(B) The average ROC AUC performance of the AggMapNet models that are trained on the four metagenomic Disease-Set1 (cirrhosis, IBD, obesity, and T2D binary task) with the RUE, MDS, ISOMAP, LLE, t-SNE, and UMAP 2D microbiomeprint Fmaps as inputs. The model performances were evaluated by the average AUC of 10-fold cross-valuation repeated 10 times (total 100 data points), and the SD error bars of 10 repeats are shown. The paired t test (100 pairs of RUE versus each ME method) was used to test the significance of the difference between the random embedding RUE and ME performance. p values for the significant levels: ∗∗∗∗p ≤ 0.0001, ∗∗∗0.0001 < p ≤ 0.001, ∗∗0.001 < p ≤ 0.01, ∗0.01 < p ≤ 0.05; not significant (n.s.), 0.05 < p ≤ 1.

Developments history of AI models. Model structures improved from simple to integrated and complex (from decision tree to boosting series algorithm, from MLP to transformer), feature extraction/engineering improved from manual to automated (from traditional machine learning to deep learning), and application fields from limited to wide. Various applications emerged with the improvements of AI and a brief list of their corresponding networks are showed in the right side of figure. Full names of the abbreviations: Adaline: Adaptive Linear Element; KNN: K‐Nearest Neighbor; BP: Back Propagation; CART: Classification and Regression Tree; SVM: Support Vector Machine; LASSO: Least Absolute Shrinkage and Selection Operator; KPCA: Kernel Principal Component Analysis; GBM: Gradient Boosting Machine; LLE: Locally Linear Embedding; ISOMAP: Isometric Feature Mapping; t‐SNE: t‐distributed Stochastic Neighbor Embedding; XGBoost: eXtreme Gradient Boosting; MLP: Multi‐Layer Perceptron; LSTM: Long Short‐Term Memory; DBN: Deep Belief Net; VAE: Variational Autoencoder; GAN: Generative Adversarial Network; ResNet: Residual Neural Network; DenseNet: Dense Convolutional Network; FCN: Fully Convolutional Network; ViT: Vision Transformer; SENet: Squeeze‐and‐Excitation Network; Faster RCNN: Faster Region‐based Convolutional Network. AI, artificial intelligence.