Nonlinear dimensionality reduction lacks interpretability due to the absence of source features in low-dimensional embedding space. We propose an interpretable method featMAP to preserve source features by tangent space embedding. The core of our proposal is to utilize local singular value decomposition (SVD) to approximate the tangent space which is embedded to low-dimensional space by maintaining the alignment. Based on the embedding tangent space, featMAP enables the interpretability by locally demonstrating the source features and feature importance. Furthermore, featMAP embeds the data points by anisotropic projection to preserve the local similarity and original density. We apply featMAP to interpreting digit classification, object detection and MNIST adversarial examples. FeatMAP uses source features to explicitly distinguish the digits and objects and to explain the misclassification of adversarial examples. We also compare featMAP with other state-of-the-art methods on local and global metrics.
This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation of Laplacian. Then, we cover the cuts of graph and spectral clustering which applies clustering in a subspace of data. Different optimization variants of Laplacian eigenmap and its out-of-sample extension are explained. Thereafter, we introduce the locality preserving projection and its kernel variant as linear special cases of Laplacian eigenmap. Versions of…
Explicitly or implicitly, most of dimensionality reduction methods need to determine which samples are neighbors and the similarity between the neighbors in the original highdimensional space. The projection matrix is then learned on the assumption that the neighborhood information (e.g., the similarity) is known and fixed prior to learning. However, it is difficult to precisely measure the intrinsic similarity of samples in high-dimensional space because of the curse of dimensionality. Consequently, the neighbors selected according to such similarity might and the projection matrix obtained a…
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
When performing classification tasks, raw high dimensional features often contain redundant information, and lead to increased computational complexity and overfitting. In this paper, we assume the data samples lie on a single underlying smooth manifold, and define intra-class and inter-class similarities using pairwise local kernel distances. We aim to find a linear projection to maximize the intra-class similarities and minimize the inter-class similarities simultaneously, so that the projected low dimensional data has optimized pairwise distances based on the label information, which is mor…
In ordinary Dimensionality Reduction (DR), each data instance in a high dimensional space (original space), or on a distance matrix denoting original space distances, is mapped to (projected onto) one point in a low dimensional space (visual space), building a layout of projected points trying to preserve as much as possible some property of data such as distances, neighbourhood relationships, and/or topology structures, with the ultimate goal of approximating semantic properties of data with preserved geometric properties or topology structures in visual space. In this paper, the concept of M…
University of Cambridge（剑桥大学）；Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）
Recent advances in machine learning allow us to analyze and describe the content of high-dimensional data like text, audio, images or other signals. In order to visualize that data in 2D or 3D, usually Dimensionality Reduction (DR) techniques are employed. Most of these techniques, e.g., PCA or t-SNE, produce static projections without taking into account corrections from humans or other data exploration scenarios. In this work, we propose the interactive Similarity Projection (iSP), a novel interactive DR framework based on similarity embeddings, where we form a differentiable objective based…
Principal Component Analysis (PCA) is known to be the most widely applied dimensionality reduction approach. A lot of improvements have been done on the traditional PCA, in order to obtain optimal results in the dimensionality reduction of various datasets. In this paper, we present an improvement to the traditional PCA approach called Multiplicative factoring Principal Component Analysis (MPCA). The advantage of MPCA over the traditional PCA is that a penalty is imposed on the occurrence space through a multiplier to make negligible the effect of outliers in seeking out projections. Here we a…
University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
This paper introduces the `Projectron' as a new neural network architecture that uses Radon projections to both classify and represent medical images. The motivation is to build shallow networks which are more interpretable in the medical imaging domain. Radon transform is an established technique that can reconstruct images from parallel projections. The Projectron first applies global Radon transform to each image using equidistant angles and then feeds these transformations for encoding to a single layer of neurons followed by a layer of suitable kernels to facilitate a linear separation of…
Linear discriminant analysis (LDA) is a fundamental method for feature extraction and dimensionality reduction. Despite having many variants, classical LDA has its own importance, as it is a keystone in human knowledge about statistical pattern recognition. For a dataset containing C clusters, the classical solution to LDA extracts at most C-1 features. Here, we introduce a novel solution to classical LDA, called LDA++, that yields C features, each interpretable as measuring similarity to one cluster. This novel solution bridges dimensionality reduction and multiclass classification. Specifica…
CLIP-PAE: Projection-Augmentation Embedding to Extract Relevant Features for a Disentangled, Interpretable, and Controllable Text-Guided Face Manipulation
Dimensionality reduction is a crucial preprocessing for hyperspectral data analysis - finding an appropriate subspace is often required for subsequent image classification. In recent work, we proposed supervised angular information based dimensionality reduction methods to find effective subspaces. Since unlabeled data are often more readily available compared to labeled data, we propose an unsupervised projection that finds a lower dimensional subspace where local angular information is preserved. To exploit spatial information from the hyperspectral images, we further extend our unsupervised…
Erasmus MC（伊拉斯姆斯大学医学中心）；Delft University of Technology（代尔夫特理工大学）
Confounding bias is a crucial problem when applying machine learning to practice, especially in clinical practice. We consider the problem of learning representations independent to multiple biases. In literature, this is mostly solved by purging the bias information from learned representations. We however expect this strategy to harm the diversity of information in the representation, and thus limiting its prospective usage (e.g., interpretation). Therefore, we propose to mitigate the bias while keeping almost all information in the latent representations, which enables us to observe and int…
Bernoulli Institute/Department of Computer Science, University of Groningen（格罗宁根大学伯努利学院/计算机科学系）；University of Groningen（格罗宁根大学）；Bernoulli Institute（伯努利学院）；Department of Computer Science（计算机科学系）；Groningen（格罗宁根）
Sharpened dimensionality reduction (SDR), which belongs to the class of multidimensional projection techniques, has recently been introduced to tackle the challenges in the exploratory and visual analysis of high-dimensional data. SDR has been applied to various real-world datasets, such as human activity sensory data and astronomical datasets. However, manually labeling the samples from the generated projection are expensive. To address this problem, we propose here to use clustering methods such as k-means, Hierarchical Clustering, Density-Based Spatial Clustering of Applications with Noise …
A fundamental question in many data analysis settings is the problem of discerning the "natural" dimension of a data set. That is, when a data set is drawn from a manifold (possibly with noise), a meaningful aspect of the data is the dimension of that manifold. Various approaches exist for estimating this dimension, such as the method of Secant-Avoidance Projection (SAP). Intuitively, the SAP algorithm seeks to determine a projection which best preserves the lengths of all secants between points in a data set; by applying the algorithm to find the best projections to vector spaces of various d…
Australian National University（澳大利亚国立大学）；NICTA（国家信息与通信技术研究院）
Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices --especially of high-dimensional ones-- comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD…
Beijing University of Technology（北京工业大学）；The University of Sydney（悉尼大学）；Dalian University of Technology（大连理工大学）
Learning on Grassmann manifold has become popular in many computer vision tasks, with the strong capability to extract discriminative information for imagesets and videos. However, such learning algorithms particularly on high-dimensional Grassmann manifold always involve with significantly high computational cost, which seriously limits the applicability of learning on Grassmann manifold in more wide areas. In this research, we propose an unsupervised dimensionality reduction algorithm on Grassmann manifold based on the Locality Preserving Projections (LPP) criterion. LPP is a commonly used d…
Australian National University（澳大利亚国立大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the h…
University of Maryland（马里兰大学）；Rutgers University（罗格斯大学）
It is proven that encoding images and videos through Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, can lead to increased classification performance. Taking into account manifold geometry is typically done via embedding the manifolds in tangent spaces, or Reproducing Kernel Hilbert Spaces (RKHS). Recently, it was shown that embedding such manifolds into a Random Projection Spaces (RPS), rather than RKHS or tangent space, leads to higher classification and clustering performance. However, based on structure and dimensionality of the r…
The approximation of nonlinear kernels via linear feature maps has recently gained interest due to their applications in reducing the training and testing time of kernel-based learning algorithms. Current random projection methods avoid the curse of dimensionality by embedding the nonlinear feature space into a low dimensional Euclidean space to create nonlinear kernels. We introduce a Layered Random Projection (LaRP) framework, where we model the linear kernels and nonlinearity separately for increased training efficiency. The proposed LaRP framework was assessed using the MNIST hand-written …
As a typical dimensionality reduction technique, random projection can be simply implemented with linear projection, while maintaining the pairwise distances of high-dimensional data with high probability. Considering this technique is mainly exploited for the task of classification, this paper is developed to study the construction of random matrix from the viewpoint of feature selection, rather than of traditional distance preservation. This yields a somewhat surprising theoretical result, that is, the sparse random matrix with exactly one nonzero element per column, can present better featu…
Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology.…
Dimensionality-reduction methods are a fundamental tool in the analysis of large data sets. These algorithms work on the assumption that the "intrinsic dimension" of the data is generally much smaller than the ambient dimension in which it is collected. Alongside their usual purpose of mapping data into a smaller dimension with minimal information loss, dimensionality-reduction techniques implicitly or explicitly provide information about the dimension of the data set. In this paper, we propose a new statistic that we call the $κ$-profile for analysis of large data sets. The $κ$-profile arises…
Shenyang Institute of Automation, Chinese Academy of Sciences（中国科学院沈阳自动化研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Key Laboratory of Optical-Electronics Information Processing（光电信息处理重点实验室）
This paper proposes a generalized framework with joint normalization which learns lower-dimensional subspaces with maximum discriminative power by making use of the Riemannian geometry. In particular, we model the similarity/dissimilarity between subspaces using various metrics defined on Grassmannian and formulate dimen-sionality reduction as a non-linear constraint optimization problem considering the orthogonalization. To obtain the linear mapping, we derive the components required to per-form Riemannian optimization (e.g., Riemannian conju-gate gradient) from the original Grassmannian thro…
Plant species identification is time consuming, costly, and requires lots of efforts, and expertise knowledge. In recent, many researchers use deep learning methods to classify plants directly using plant images. While deep learning models have achieved a great success, the lack of interpretability limit their widespread application. To overcome this, we explore the use of interpretable, measurable and computer-aided features extracted from plant leaf images. Image processing is one of the most challenging, and crucial steps in feature-extraction. The purpose of image processing is to improve …
National University of Sciences and Technology（国立科技大学）；University of Insubria（因苏布里亚大学）
Majority of the current dimensionality reduction or retrieval techniques rely on embedding the learned feature representations onto a computable metric space. Once the learned features are mapped, a distance metric aids the bridging of gaps between similar instances. Since the scaled projection is not exploited in these methods, discriminative embedding onto a hyperspace becomes a challenge. In this paper, we propose to inwardly scale feature representations in proportional to projecting them onto a hypersphere manifold for discriminative analysis. We further propose a novel, yet simpler, conv…
University of Bari “A. Moro”（巴里大学“A. Moro”）；Université de Mons（蒙斯大学）
Linear dimensionality reduction techniques are powerful tools for image analysis as they allow the identification of important features in a data set. In particular, nonnegative matrix factorization (NMF) has become very popular as it is able to extract sparse, localized and easily interpretable features by imposing an additive combination of nonnegative basis elements. Nonnegative matrix underapproximation (NMU) is a closely related technique that has the advantage to identify features sequentially. In this paper, we propose a variant of NMU that is particularly well suited for image analysis…
