Manifold Fitting Research Articles

Retraction Note: Manifold fitting algorithm of noisy manifold data based on variable-scale spectral graph

Retraction Note: Manifold fitting algorithm of noisy manifold data based on variable-scale spectral graph

Single-cell analysis via manifold fitting: A framework for RNA clustering and beyond

Single-cell RNA sequencing (scRNA-seq) data, susceptible to noise arising from biological variability and technical errors, can distort gene expression analysis and impact cell similarity assessments, particularly in heterogeneous populations. Current methods, including deep learning approaches, often struggle to accurately characterize cell relationships due to this inherent noise. To address these challenges, we introduce scAMF (Single-cell Analysis via Manifold Fitting), a framework designed to enhance clustering accuracy and data visualization in scRNA-seq studies. At the heart of scAMF lies the manifold fitting module, which effectively denoises scRNA-seq data by unfolding their distribution in the ambient space. This unfolding aligns the gene expression vector of each cell more closely with its underlying structure, bringing it spatially closer to other cells of the same cell type. To comprehensively assess the impact of scAMF, we compile a collection of 25 publicly available scRNA-seq datasets spanning various sequencing platforms, species, and organ types, forming an extensive RNA data bank. In our comparative studies, benchmarking scAMF against existing scRNA-seq analysis algorithms in this data bank, we consistently observe that scAMF outperforms in terms of clustering efficiency and data visualization clarity. Further experimental analysis reveals that this enhanced performance stems from scAMF's ability to improve the spatial distribution of the data and capture class-consistent neighborhoods. These findings underscore the promising application potential of manifold fitting as a tool in scRNA-seq analysis, signaling a significant enhancement in the precision and reliability of data interpretation in this critical field of study.

Manifold fitting with CycleGAN

Manifold fitting, which offers substantial potential for efficient and accurate modeling, poses a critical challenge in nonlinear data analysis. This study presents an approach that employs neural networks to fit the latent manifold. Leveraging the generative adversarial framework, this method learns smooth mappings between low-dimensional latent space and high-dimensional ambient space, echoing the Riemannian exponential and logarithmic maps. The well-trained neural networks provide estimations for the latent manifold, facilitate data projection onto the manifold, and even generate data points that reside directly within the manifold. Through an extensive series of simulation studies and real data experiments, we demonstrate the effectiveness and accuracy of our approach in capturing the inherent structure of the underlying manifold within the ambient space data. Notably, our method exceeds the computational efficiency limitations of previous approaches and offers control over the dimensionality and smoothness of the resulting manifold. This advancement holds significant potential in the fields of statistics and computer science. The seamless integration of powerful neural network architectures with generative adversarial techniques unlocks possibilities for manifold fitting, thereby enhancing data analysis. The implications of our findings span diverse applications, from dimensionality reduction and data visualization to generating authentic data. Collectively, our research paves the way for future advancements in nonlinear data analysis and offers a beacon for subsequent scholarly pursuits.

RETRACTED ARTICLE: Manifold fitting algorithm of noisy manifold data based on variable-scale spectral graph

RETRACTED ARTICLE: Manifold fitting algorithm of noisy manifold data based on variable-scale spectral graph

Multi-View Representation Learning via Dual Optimal Transportation

Recently, multi-view representation learning has gained rapid growth in various fields. Most of previous multi-view learning methods rely on strong notions of distances that often provide no useful gradients in deep network training, which greatly degrades the performance in merging complementary information of views. To address this challenge, a multi-view representation learning network with dual optimal transportation (MDOT-Net) is proposed to capture fusion representations embedded in a common manifold with the weak topology. In MDOT-Net, the multi-view representation learning is modelled as an optimal transportation (OT) problem in manifold fitting, which is further decomposed into the intra-view OT and the inter-view OT. The intra-view OT is implemented by a view-specific adversarial variational network, which accurately captures local manifold structures within views by leveraging fusion knowledge. The inter-view OT is implemented by a view-fusion adversarial inference network, which models fusion representations compatible with diversities of sub-manifolds by utilizing view-specific knowledge. The two OTs boost mutually by providing prior knowledge to each other in multi-view representation learning. Finally, numerous experiments are conducted on four benchmark datasets, and the results demonstrate MDOT-Net achieves the state-of-the-art performance.

A Novel Approach to Algebraic Fitting of a Pencil of Quadrics for Planar 4R Motion Synthesis

The use of the image space of planar displacements for planar motion approximation is a well studied subject. While the constraint manifolds associated with planar four-bar linkages are algebraic, geometric (or normal) distances have been used as default metric for nonlinear least squares fitting of these algebraic manifolds. This paper presents a new formulation for the manifold fitting problem using algebraic distance and shows that the problem can be solved by fitting a pencil of quadrics with linear coefficients to a set of image points of a given set of displacements. This linear formulation leads to a simple and fast algorithm for kinematic synthesis in the image space.

Morphological appearance manifolds for group-wise morphometric analysis

Computational anatomy quantifies anatomical shape based on diffeomorphic transformations of a template. However, different templates warping algorithms, regularization parameters, or templates, lead to different representations of the same exact anatomy, raising a uniqueness issue: variations of these parameters are confounding factors as they give rise to non-unique representations. Recently, it has been shown that learning the equivalence class derived from the multitude of representations of a given anatomy can lead to improved and more stable morphological descriptors. Herein, we follow that approach, by approximating this equivalence class of morphological descriptors by a (nonlinear) morphological appearance manifold fitting to the data via a locally linear model. Our approach parallels work in the computer vision field, in which variations lighting, pose and other parameters lead to image appearance manifolds representing the exact same figure in different ways. The proposed framework is then used for group-wise registration and statistical analysis of biomedical images, by employing a minimum variance criterion to perform manifold-constrained optimization, i.e. to traverse each individual's morphological appearance manifold until group variance is minimal. The hypothesis is that this process is likely to reduce aforementioned confounding effects and potentially lead to morphological representations reflecting purely biological variations, instead of variations introduced by modeling assumptions and parameter settings.

Optimized method for computing (18)O/(16)O ratios of differentially stable-isotope labeled peptides in the context of postdigestion (18)O exchange/labeling.

Differential (18)O/(16)O stable isotope labeling of peptides that relies on enzyme-catalyzed oxygen exchange at their carboxyl termini in the presence of H(2)(18)O has been widely used for relative quantitation of peptides/proteins. The role of tryptic proteolysis in bottom-up shotgun proteomics and low reagent costs have made trypsin-catalyzed (18)O postdigestion exchange a convenient and affordable stable isotope labeling approach. However, it is known that trypsin-catalyzed (18)O exchange at the carboxyl terminus is in many instances inhomogeneous/incomplete. The extent of the (18)O exchange/incorporation fluctuates from peptide to peptide mostly due to variable enzyme-substrate affinity. Thus, accurate calculation and interpretation of peptide ratios are analytically complicated and in some regard deficient. Therefore, a computational approach capable of improved measurement of actual (18)O incorporation for each differentially labeled peptide pair is needed. In this regard, we have developed an algorithmic method that relies on the trapezoidal rule to integrate peak intensities of all detected isotopic species across a particular peptide ion over the retention time, which fits the isotopic manifold to Poisson distributions. Optimal values for manifold fitting were calculated and then (18)O/(16)O ratios derived via evolutionary programming. The algorithm is tested using trypsin-catalyzed (18)O postdigestion exchange to differentially label bovine serum albumin (BSA) at a priori determined ratios. Both accuracy and precision are improved utilizing this rigorous mathematical approach. We further demonstrate the effectiveness of this method to accurately calculate (18)O/(16)O ratios in a large scale proteomic quantitation of detergent resistant membrane microdomains (DRMMs) isolated from cells expressing wild-type HIV-1 Gag and its nonmyristylated mutant.

Motion Recovery by Integrating over the Joint Image Manifold

Recovery of epipolar geometry is a fundamental problem in computer vision. The introduction of the joint image (JIM) allows to treat the recovery of camera motion and epipolar geometry as the problem of fitting a manifold to the data measured in a stereo pair. The manifold has a singularity and boundary, therefore special care must be taken when fitting it. Four fitting methods are discussed--direct, algebraic, geometric, and the integrated maximum likelihood (IML) based method. The first three methods are the exact analogues of three common methods for recovering epipolar geometry. The more recently introduced IML method seeks the manifold which has the highest support, in the sense that the largest measure of its points are close to the data. While computationally more intensive than the other methods, its results are better in some scenarios. Both simulations and experiments suggest that the advantages of IML manifold fitting carry over to the task of recovering epipolar geometry, especially when the extent of the data and/or the motion are small.

Bayesian Model Estimation and Selection for Epipolar Geometry and Generic Manifold Fitting

Computer vision often involves estimating models from visual input. Sometimes it is possible to fit several different models or hypotheses to a set of data, and a decision must be made as to which is most appropriate. This paper explores ways of automating the model selection process with specific emphasis on the least squares problem of fitting manifolds (in particular algebraic varieties e.g. lines, algebraic curves, planes etc.) to data points, illustrated with respect to epipolar geometry. The approach is Bayesian and the contribution three fold, first a new Bayesian description of the problem is laid out that supersedes the author's previous maximum likelihood formulations, this formulation will reveal some hidden elements of the problem. Second an algorithm, ‘MAPSAC’, is provided to obtain the robust MAP estimate of an arbitrary manifold. Third, a Bayesian model selection paradigm is proposed, the Bayesian formulation of the manifold fitting problem uncovers an elegant solution to this problem, for which a new method ‘GRIC’ for approximating the posterior probability of each putative model is derived. This approximations bears some similarity to the penalized likelihoods used by AIC, BIC and MDL however it is far more accurate in situations involving large numbers of latent variables whose number increases with the data. This will be empirically and theoretically demonstrated.

Geometric data fitting

Given a dense set of points lying on or near an embedded submanifold M0 ⊂ ℝn of Euclidean space, the manifold fitting problem is to find an embedding F : M → ℝn that approximates M0 in the sense of least squares. When the dataset is modeled by a probability distribution, the fitting problem reduces to that of finding an embedding that minimizes Ed[F], the expected square of the distance from a point in ℝn to F(M). It is shown that this approach to the fitting problem is guaranteed to fail because the functional Ed has no local minima. This problem is addressed by adding a small multiple k of the harmonic energy functional to the expected square of the distance. Techniques from the calculus of variations are then used to study this modified functional.

Extremal Properties of Principal Embeddings

Given a dense set of points lying on or near an embedded submanifold M 0⊂ R n of Euclidean space, the manifold fitting problem is to find an embedding F :M→ R n that approximates M 0 in the sense of least squares. When the dataset is modeled by a probability distribution, the fitting problem reduces to that of finding an embedding that minimizes E d[F] , the expected square of the distance from a point in R n to F(M). In this article it is shown that this approach to the fitting problem is guaranteed to fail because the functional E d has no local minima. This explains why cross-validation does not appear to be a viable method for choosing the complexity of principal manifold estimates.