vMF Distribution Research Articles

Probabilistic Contrastive Learning for Long-Tailed Visual Recognition.

Long-tailed distributions frequently emerge in real-world data, where a large number of minority categories contain a limited number of samples. Such imbalance issue considerably impairs the performance of standard supervised learning algorithms, which are mainly designed for balanced training sets. Recent investigations have revealed that supervised contrastive learning exhibits promising potential in alleviating the data imbalance. However, the performance of supervised contrastive learning is plagued by an inherent challenge: it necessitates sufficiently large batches of training data to construct contrastive pairs that cover all categories, yet this requirement is difficult to meet in the context of class-imbalanced data. To overcome this obstacle, we propose a novel probabilistic contrastive (ProCo) learning algorithm that estimates the data distribution of the samples from each class in the feature space, and samples contrastive pairs accordingly. In fact, estimating the distributions of all classes using features in a small batch, particularly for imbalanced data, is not feasible. Our key idea is to introduce a reasonable and simple assumption that the normalized features in contrastive learning follow a mixture of von Mises-Fisher (vMF) distributions on unit space, which brings two-fold benefits. First, the distribution parameters can be estimated using only the first sample moment, which can be efficiently computed in an online manner across different batches. Second, based on the estimated distribution, the vMF distribution allows us to sample an infinite number of contrastive pairs and derive a closed form of the expected contrastive loss for efficient optimization. Other than long-tailed problems, ProCo can be directly applied to semi-supervised learning by generating pseudo-labels for unlabeled data, which can subsequently be utilized to estimate the distribution of the samples inversely. Theoretically, we analyze the error bound of ProCo. Empirically, extensive experimental results on supervised/semi-supervised visual recognition and object detection tasks demonstrate that ProCo consistently outperforms existing methods across various datasets.

On regime changes in text data using hidden Markov model of contaminated vMF distribution

On regime changes in text data using hidden Markov model of contaminated vMF distribution

Statistical insights of polarization speckle via von Mises–Fisher distribution on the Poincaré sphere

Polarization speckles generated via random scattering of light are ubiquitous in natural and engineered systems. They not only manifest intensity fluctuations but also reveal a spatially fluctuating, random polarization distribution. The precise morphology of the polarization speckle pattern serves as a deterministic signature of the light’s state of polarization fluctuation within a scattering medium. Given the inherent randomness of polarization speckle patterns, a statistical approach emerges as the most pragmatic method for their analysis. Stokes parameters, implemented as temporal or spatial averages, are utilized for this purpose. However, within a polarization speckle field featuring a specific spatial average of Stokes parameters, the polarization state exhibits spatial variations across the speckle pattern. These random polarization fluctuations can be effectively modeled using a particular probability density function (PDF), visually represented on the Poincaré sphere. In this work, von Mises–Fisher (vMF) distribution on the Poincaré sphere is extended and applied to demonstrate a statistical insight of polarization speckle fields. A complete theoretical basis is established to investigate the spatial fluctuation of the state of polarization in the polarization speckle using vMF distribution on the Poincaré sphere, including the spatial mean direction, and spatial concentration parameter. Behavior of the marginal vMF distribution on the axes of the Poincaré sphere and its association with the probability density function of the normalized at-the-point Stokes parameters for three different polarization speckles are examined by experiment and simulation. The experimental results are in good agreement with the simulation results and confirm the usefulness of the established theoretical framework for the analysis of the polarization speckles. Characterization of spatial polarization fluctuation offers significant applications, such as in polarimetric analysis and optical sensing, and the same analogy can be used in quantum optics.

Data-Free Generalized Zero-Shot Learning

Deep learning models have the ability to extract rich knowledge from large-scale datasets. However, the sharing of data has become increasingly challenging due to concerns regarding data copyright and privacy. Consequently, this hampers the effective transfer of knowledge from existing data to novel downstream tasks and concepts. Zero-shot learning (ZSL) approaches aim to recognize new classes by transferring semantic knowledge learned from base classes. However, traditional generative ZSL methods often require access to real images from base classes and rely on manually annotated attributes, which presents challenges in terms of data restrictions and model scalability. To this end, this paper tackles a challenging and practical problem dubbed as data-free zero-shot learning (DFZSL), where only the CLIP-based base classes data pre-trained classifier is available for zero-shot classification. Specifically, we propose a generic framework for DFZSL, which consists of three main components. Firstly, to recover the virtual features of the base data, we model the CLIP features of base class images as samples from a von Mises-Fisher (vMF) distribution based on the pre-trained classifier. Secondly, we leverage the text features of CLIP as low-cost semantic information and propose a feature-language prompt tuning (FLPT) method to further align the virtual image features and textual features. Thirdly, we train a conditional generative model using the well-aligned virtual image features and corresponding semantic text features, enabling the generation of new classes features and achieve better zero-shot generalization. Our framework has been evaluated on five commonly used benchmarks for generalized ZSL, as well as 11 benchmarks for the base-to-new ZSL. The results demonstrate the superiority and effectiveness of our approach. Our code is available in https://github.com/ylong4/DFZSL.

Probabilistic Keyphrase Generation From Copy and Generating Spaces.

Keyphrase generation is one of the most fundamental tasks in natural language processing (NLP). Most existing works on keyphrase generation mainly focus on using holistic distribution to optimize the negative log-likelihood loss, but they do not directly manipulate the copy and generating spaces, which may reduce the generability of the decoder. Additionally, existing keyphrase models are either unable to determine the dynamic numbers of keyphrases or produce the number of keyphrases implicitly. In this article, we propose a probabilistic keyphrase generation model from copy and generating spaces. The proposed model is built upon the vanilla variational encoder-decoder (VED) framework. On top of VED, two separate latent variables are adopted to model the distribution of data within the latent copy and generating spaces, respectively. Specifically, we adopt a von Mises-Fisher (vMF) distribution to obtain a condensed variable for modifying the generating probability distribution over the predefined vocabulary. Meanwhile, we utilize a clustering module, which is designed to promote Gaussian Mixture learning and subsequently extract a latent variable for the copy probability distribution. Moreover, we utilize a natural property of the Gaussian mixture network and use the number of filtered components to determine the number of keyphrases. The approach is trained based on latent variable probabilistic modeling, neural variational inference, and self-supervised learning. Experiments on social media and scientific article datasets outperform the state-of-the-art baselines in generating accurate predictions and controllable keyphrase numbers.

Von Mises-Fisher Elliptical Distribution.

Modern probabilistic learning systems mainly assume symmetric distributions, however, real-world data typically obey skewed distributions and are thus not adequately modeled through symmetric distributions. To address this issue, a generalization of symmetric distributions called elliptical distributions are increasingly used, together with further improvements based on skewed elliptical distributions. However, existing approaches are either hard to estimate or have complicated and abstract representations. To this end, we propose a novel approach based on the von-Mises-Fisher (vMF) distribution to obtain an explicit and simple probability representation of skewed elliptical distributions. The analysis shows that this not only allows us to design and implement nonsymmetric learning systems but also provides a physically meaningful and intuitive way of generalizing skewed distributions. For rigor, the proposed framework is proven to share important and desirable properties with its symmetric counterpart. The proposed vMF distribution is demonstrated to be easy to generate and stable to estimate, both theoretically and through examples.

Modeling the Free Inclinations of the Classical Kuiper Belt with the von Mises–Fisher Distribution

The orientations of orbital planes of minor planets are directional random variables. Their free inclination is the deviation of the orbit plane from the plane forced by the major planets. We construct a model of the distribution of free inclinations of classical Kuiper Belt objects (CKBOs) based on the von Mises–Fisher (vMF) distribution function, the analog of the normal distribution for directional statistics. The CKBOs are known to have a “cold” component of orbit planes concentrated near the forced plane and a more widely dispersed “hot” component. Adopting a model with a linear combination of two vMF functions, we find that the cold and hot components account for 57% and 43%, characterized by widths of 1.°7 and 12.°9, respectively. This model improves upon previous models based on smaller observational samples and empirical choices of functional forms for inclination distributions.

Markov Chain Mixture Models for Real‐Time Direct Illumination

AbstractWe present a novel technique to efficiently render complex direct illumination in real‐time. It is based on a spatio‐temporal randomized mixture model of von Mises‐Fisher (vMF) distributions in screen space. For every pixel we determine the vMF distribution to sample from using a Markov chain process which is targeted to capture important features of the integrand. By this we avoid the storage overhead of finite‐component deterministic mixture models, for which, in addition, determining the optimal component count is challenging. We use stochastic multiple importance sampling (SMIS) to be independent of the equilibrium distribution of our Markov chain process, since it cancels out in the estimator. Further, we use the same sample to advance the Markov chain and to construct the SMIS estimator and local Markov chain state permutations avoid the resulting bias due to dependent sampling. As a consequence we require one ray per sample and pixel only. We evaluate our technique using implementations in a research renderer as well as a classic game engine with highly dynamic content. Our results show that it is efficient and quickly readapts to dynamic conditions. We compare to spatio‐temporal resampling (ReSTIR), which can suffer from correlation artifacts due to its non‐adapting candidate distributions that can deviate strongly from the integrand. While we focus on direct illumination, our approach is more widely applicable and we exemplarily show the rendering of caustics.

A Robust Pedestrian Re-Identification and Out-Of-Distribution Detection Framework

Pedestrian re-identification is an important field due to its applications in security and safety. Most current solutions for this problem use CNN-based feature extraction and assume that only the identities that are in the training data can be recognized. On the one hand, the pedestrians in the training data are called In-Distribution (ID). On the other hand, in real-world scenarios, new pedestrians and objects can appear in the scene, and the model should detect them as Out-Of-Distribution (OOD). In our previous study, we proposed a pedestrian re-identification based on von Mises–Fisher (vMF) distribution. Each identity is embedded in the unit sphere as a compact vMF distribution far from other identity distributions. Recently, a framework called Virtual Outlier Synthetic (VOS) was proposed, which detects OOD based on synthesizing virtual outliers in the embedding space in an online manner. Their approach assumes that the samples from the same object map to a compact space, which aligns with the vMF-based approach. Therefore, in this paper, we revisited the vMF approach and merged it with VOS to detect OOD data points. Experiment results showed that our framework was able to detect new pedestrians that do not exist in the training data in the inference phase. Furthermore, this framework improved the re-identification performance and holds a significant potential in real-world scenarios.

A von Mises–Fisher distribution for the orbital poles of the plutinos

ABSTRACT Small Solar system bodies have widely dispersed orbital poles, posing challenges to dynamical models of Solar system origin and evolution. To characterize the orbit pole distribution of dynamical groups of small bodies it helps to have a functional form for a model of the distribution function. Previous studies have used the small-inclination approximation and adopted variations of the normal distribution to model orbital inclination dispersions. Because the orbital pole is a directional variable, its distribution can be more appropriately modelled with directional statistics. We describe the von Mises–Fisher (vMF) distribution on the surface of the unit sphere for application to small bodies’ orbital poles. We apply it to the orbit pole distribution of the observed Plutinos. We find a mean pole located at inclination i0 = 3.57° and longitude of ascending node Ω0 = 124.38° (in the J2000 reference frame), with a 99.7 per cent confidence cone of half-angle 1.68°. We also estimate a debiased mean pole located 4.6° away, at i0 = 2.26°, Ω0 = 292.69°, of similar-size confidence cone. The vMF concentration parameter of Plutino inclinations (relative to either mean pole estimate) is κ = 31.6. This resembles a Rayleigh distribution function, with width parameter σ = 10.2°. Unlike previous models, the vMF model naturally accommodates all physical inclinations (and no others), whereas Rayleigh or Gaussian models must be truncated to the physical inclination range 0–180°. Further work is needed to produce a theory for the mean pole of the Plutinos against which to compare the observational results.

A new distance-based distribution: Detecting concentration in directional data

In this article, we propose a simple method to study high concentrations in spherical data, in particular in the directional perspective. To this end, we define a distance-based distribution in the interval (0,1), called the T-statistic distance (DT) law, to describe the scattering of points on the unit sphere. Our model is derived from the von Mises–Fisher (vMF) distribution, which is one of the most known directional laws. We show that if the data are vMF distributed, their concentration can be modeled by our distribution. Some of its properties are derived and discussed: Moment generating function, kurtosis and skewness. Likelihood-based inference procedures are provided for both points and hypotheses concerning the DT concentration. Further, we propose a new test statistic in terms of the DT distribution to deal with scattering within spherical data and derive its exact density. Numerical studies show that maximum likelihood estimates behave asymptotically well even for small sample sizes and that the likelihood ratio test for the DT distribution often performs better than Wald tests. We apply our model to paleomagnetic data to illustrate how it is used to analyze spherical data concentration. Results show that the distance-based approach works well to identify high concentration on the unit sphere.

Correlation between speed and turning naturally arises for sparsely sampled cell movements

Mechanisms regulating cell movement are not fully understood. One feature of cell movement that determines how far cells displace from an initial position is persistence, the ability to perform movements in a direction similar to the previous movement direction. Persistence is thus determined by turning angles (TA) between two sequential displacements and can be characterized by an average TA or persistence time. Recent studies documenting T cell movement in zebrafish found that a cell’s average speed and average TA are negatively correlated, suggesting a fundamental cell-intrinsic program whereby cells with a lower TA (and larger persistence time) are intrinsically faster (or faster cells turn less). In this paper we confirm the existence of the correlation between turning and speed for six different datasets on 3D movement of CD8 T cells in murine lymph nodes or liver. Interestingly, the negative correlation between TA and speed was observed in experiments in which liver-localized CD8 T cells rapidly displace due to floating with the blood flow, suggesting that other mechanisms besides cell-intrinsic program may be at play. By simulating correlated random walks using two different frameworks (one based on the von Mises–Fisher (vMF) distribution and another based on the Ornstein–Uhlenbeck (OU) process) we show that the negative correlation between speed and turning naturally arises when cell trajectories are sub-sampled, i.e. when the frequency of sampling is lower than frequency at which cells typically make movements. This effect is strongest when the sampling frequency is of the order of magnitude of the inverse of persistence time of cells and when cells vary in persistence time. The effect arises in part due to the sensitivity of estimated cell speeds to the frequency of imaging whereby less frequent imaging results in slower speeds. Interestingly, by using estimated persistence times for cells in two of our datasets and simulating cell movements using the OU process, we could partially reproduce the experimentally observed correlation between TA and speed without a cell-intrinsic program linking the two processes. Our results thus suggest that sub-sampling may contribute to (and perhaps fully explains) the observed correlation between speed and turning at least for some cell trajectory data and emphasize the role of sampling frequency in the inference of critical cellular parameters of cell motility such as speeds.

Parameter estimation and model-based clustering with spherical normal distribution on the unit hypersphere

In directional statistics, the von Mises-Fisher (vMF) distribution is one of the most basic and popular probability distributions for data on the unit hypersphere. Recently, the spherical normal (SN) distribution was proposed as an intrinsic counterpart to the vMF distribution by replacing the standard Euclidean norm with the great-circle distance, which is length of the shortest path joining two points on the unit sphere. Focusing on an isotropic version of SN distribution, it is shown that maximum likelihood estimators uniquely exist under mild support conditions. Since no analytic formula are available for the estimation, efficient numerical routines are proposed for parameter estimation. The estimation is considered in a general setting where non-negative weights are assigned to observations. This leads to a more interesting contribution for model-based clustering on the unit hypersphere by finite mixture model with SN distributions. Efficiency of optimization-based estimation procedures and effectiveness of SN mixture model are validated using simulated and real data examples.

Aligning 3D Curve With Surface Using Tangent and Normal Vectors for Computer-Assisted Orthopedic Surgery

Registration that aligns different views of one interested organ together is an essential technique and outstanding problem in medical robotics and image-guided surgery (IGS). This work introduces a novel rigid point set registration (PSR) approach that aims to accurately map the pre-operative space with the intra-operative space to enable successful image guidance for computer-assisted orthopaedic surgery (CAOS). The normal vectors and tangent vectors are first extracted from the pre-operative and intra-operative point sets (PSs) respectively, and are further utilized to enhance the registration accuracy and robustness. The contributions of this article are three-folds. First, we propose and formulate a novel distribution that describes the error between one normal vector and the corresponding tangent vector based on the von-Mises Fisher (vMF) distribution. Second, by modelling the anisotropic position localization error with the multi-variate Gaussian distribution, we formulate the PSR considering anisotropic localization error as a maximum likelihood estimation (MLE) problem and then solve it under the expectation maximization (EM) framework. Third, to facilitate the optimization process, the gradients of the objective function with respect to the desired parameters are computed and presented. Extensive experimental results on the human femur and pelvis models verify that the proposed approach outperforms the state-of-the-art methods, and demonstrate potential clinical values for relevant surgical navigation applications.

Geodesic projection of the von Mises–Fisher distribution for projection pursuit of directional data

We investigate geodesic projections of von Mises–Fisher (vMF) distributed directional data. The vMF distribution for random directions on the (p−1)-dimensional unit hypersphere Sp−1⊂Rp plays the role of multivariate normal distribution in directional statistics. For one-dimensional circle S1, the vMF distribution is called von Mises (vM) distribution. Projections onto geodesics are one of main ingredients of modeling and exploring directional data. We show that the projection of vMF distributed random directions onto any geodesic is approximately vM-distributed, albeit not exactly the same. In particular, the distribution of the geodesic-projected score is an infinite scale mixture of vM distributions. Approximations by vM distributions are given along various asymptotic scenarios including large and small concentrations (κ→∞,κ→0), high-dimensions (p→∞), and two important cases of double-asymptotics (p,κ→∞, κ∕p→c or κ∕p→λ), to support our claim: geodesic projections of the vMF are approximately vM. As one of potential applications of the result, we contemplate a projection pursuit exploration of high-dimensional directional data. We show that in a high dimensional model almost all geodesic-projections of directional data are nearly vM, thus measures of non-vM-ness are a viable candidate for projection index.

The 8-parameter Fisher–Bingham distribution on the sphere

The Fisher–Bingham distribution ( $$\mathrm {FB}_8$$ ) is an eight-parameter family of probability density functions (PDF) on the unit sphere that, under certain conditions, reduce to spherical analogues of bivariate normal PDFs. Due to difficulties in its interpretation and estimation, applications have been mainly restricted to subclasses of $$\mathrm {FB}_8$$ , such as the Kent ( $$\mathrm {FB}_5$$ ) or von Mises–Fisher (vMF) distributions. However, these subclasses often do not adequately describe directional data that are not symmetric along great circles. The normalizing constant of $$\mathrm {FB}_8$$ can be numerically integrated, and recently Kume and Sei showed that it can be computed using an adjusted holonomic gradient method. Both approaches, however, can be computationally expensive. In this paper, I show that the normalization of $$\mathrm {FB}_8$$ can be expressed as an infinite sum consisting of hypergeometric functions, similar to that of the $$\mathrm {FB}_5$$ . This allows the normalization to be computed under summation with adequate stopping conditions. I then fit the $$\mathrm {FB}_8$$ to two datasets using a maximum-likelihood approach and show its improvements over a fit with the more restrictive $$\mathrm {FB}_5$$ distribution.

Time-sensitive clustering evolving textual data streams

Clustering a stream of text documents is an emerging subject of interest since it is widely used in analysing the content in social media and e-journals. The aim is to find a certain structure for unlabelled data based on a similarity criterion. However, few works have focused on this field and fall in this perspective, that's why a new document clustering approach adapted to a stream of text data and test it on news articles data sets is proposed. A distributed representation of words is used, and a bottom-up approach is used to represent documents as vectors on a unit hyper-sphere. The proposed approach gains its roots from the SPherical k-means (SPKM) algorithm and its underlying mixture of von-Mises Fisher (vMF) distributions. The proposed approach yields comparable results to baseline batch algorithm for stable data streams and superior results for rapidly evolving data streams.

A Spherical Hidden Markov Model for Semantics-Rich Human Mobility Modeling

We study the problem of modeling human mobility from semantic trace data, wherein each GPS record in a trace is associated with a text message that describes the user's activity. Existing methods fall short in unveiling human movement regularities for such data, because they either do not model the text data at all or suffer from text sparsity severely. We propose SHMM, a multi-modal spherical hidden Markov model for semantics-rich human mobility modeling. Under the hidden Markov assumption, SHMM models the generation process of a given trace by jointly considering the observed location, time, and text at each step of the trace. The distinguishing characteristic of SHMM is the text modeling part. We use fixed-size vector representations to encode the semantics of the text messages, and model the generation of the l2-normalized text embeddings on a unit sphere with the von Mises-Fisher (vMF) distribution. Compared with other alternatives like multi-variate Gaussian, our choice of the vMF distribution not only incurs much fewer parameters, but also better leverages the discriminative power of text embeddings in a directional metric space. The parameter inference for the vMF distribution is non-trivial since it involves functional inversion of ratios of Bessel functions. We theoretically prove, for the first time, that: 1) the classical Expectation-Maximization algorithm is able to work with vMF distributions; and 2) while closed-form solutions are hard to be obtained for the M-step, Newton's method is guaranteed to converge to the optimal solution with quadratic convergence rate. We have performed extensive experiments on both synthetic and real-life data. The results on synthetic data verify our theoretical analysis; while the results on real-life data demonstrate that SHMM learns meaningful semantics-rich mobility models, outperforms state-of-the-art mobility models for next location prediction, and incurs lower training cost.

Real‐Time Linear BRDF MIP‐Mapping

AbstractWe present a new technique to jointly MIP‐map BRDF and normal maps. Starting with generating an instant BRDF map, our technique builds its MIP‐mapped versions based on a highly efficient algorithm that interpolates von Mises‐Fisher (vMF) distributions. In our BRDF MIP‐maps, each pixel stores a vMF mixture approximating the average of all BRDF lobes from the finest level. Our method is capable of jointly MIP‐mapping BRDF and normal maps, even with high‐frequency variations, at real‐time while preserving high‐quality reflectance details. Further, it is very fast, easy to implement, and requires no precomputation.

Asymmetric biclustering with constrained von Mises-Fisher models

As a probability distribution on the high-dimensional sphere, the von Mises-Fisher (vMF) distribution is widely used for directional statistics and data analysis methods based on correlation. We consider a constrained vMF distribution for block modeling, which provides a probabilistic model of an asymmetric biclustering method that uses correlation as the similarity measure of data features. We derive the variational Bayesian inference algorithm for the mixture of the constrained vMF distributions. It is applied to a multivariate data visualization method implemented with enhanced parallel coordinate plots.