Reservation prices have proven effective in boosting revenue in Generalised Second Price (GSP) auctions, particularly in cost-per-click (CPC) settings. However, in domains like music streaming, where ads are consumed passively without user clicks, a cost-per-impression (CPM) model is more appropriate. Additionally, in the music streaming domain, user intent is typically unknown, unlike in sponsored search, making it essential to optimally leverage all available user and contextual information when setting prices. This paper addresses the challenge of optimising reservation prices in GSP auctions with CPM pricing, adopting a personalised approach that accounts for both user- and advertiser-specific factors.

Budget management is an essential capability in many online applications, including running advertising campaigns in sponsored search and allocating promotional content in recommender systems (RS). Most existing approaches to optimising budget spendings rely on improving worst-case approximation guarantees in the respective online knapsack packing problem or leveraging online learning techniques to improve an average system performance. However, worst-case approaches often underperform in practice, as extreme scenarios are uncommon, while online learning methods may lack robustness in highly non-stationary environments. In our work, we bridge these two approaches by developing an online budget pacing algorithm that preserves worst-case guarantees while improving the average allocative efficiency of budget management systems.

In-context learning is a promising approach for online policy learning of offline reinforcement learning (RL) methods, which can be achieved at inference time without gradient optimization. However, this method is hindered by significant computational costs resulting from the gathering of large training trajectory sets and the need to train large Transformer models. We address this challenge by introducing an In-context Exploration-Exploitation (ICEE) algorithm, designed to optimize the efficiency of in-context policy learning. Unlike existing models, ICEE performs an exploration-exploitation trade-off at inference time within a Transformer model, without the need for explicit Bayesian inference. Consequently, ICEE can solve Bayesian optimization problems as efficiently as Gaussian process biased methods do, but in significantly less time. Through experiments in grid world environments, we demonstrate that ICEE can learn to solve new RL tasks using only tens of episodes, marking a substantial improvement over the hundreds of episodes needed by the previous in-context learning method.

The central objective function of a variational autoencoder (VAE) is its variational lower bound (the ELBO). Here we show that for standard (i.e., Gaussian) VAEs the ELBO converges to a value given by the sum of three entropies: the (negative) entropy of the prior distribution, the expected (negative) entropy of the observable distribution, and the average entropy of the variational distributions (the latter is already part of the ELBO). Our derived analytical results are exact and apply for small as well as for intricate deep networks for encoder and decoder. Furthermore, they apply for finitely and infinitely many data points and at any stationary point (including local maxima and saddle points). The result implies that the ELBO can for standard VAEs often be computed in closed-form at stationary points while the original ELBO requires numerical approximations of integrals. As a main contribution, we provide the proof that the ELBO for VAEs is at stationary points equal to entropy sums. Numerical experiments then show that the obtained analytical results are sufficiently precise also in those vicinities of stationary points that are reached in practice. Furthermore, we discuss how the novel entropy form of the ELBO can be used to analyze and understand learning behavior. More generally, we believe that our contributions can be useful for future theoretical and practical studies on VAE learning as they provide novel information on those points in parameters space that optimization of VAEs converges to.

Users in music streaming platforms typically consume tracks sequentially in sessions by interacting with personalised playlists. To satisfy users, music platforms usually rely on recommender systems that learn users' preferences over individual tracks and rank the tracks within each playlist according to the learned preferences. However, such rankings often do not fully exploit the sequential nature of the users' consumption, which may result in a lower within-a-session consumption. In this paper, we model the sequential within-a-session preferences of users and propose an optimisation-based sequencing approach that allows for optimally incorporating such preferences into the rankings. To this end, we rely on interaction data of a major music streaming service to identify two most common aspects of the users' sequential preferences: (1) Position-Aware preferences, and (2) Local-Sequential preferences. We propose a sequencing model that can leverage each of these aspects optimally to maximise the expected total consumption from the session. We further perform an extensive offline and off-policy evaluation of our model, and carry out a large scale online randomised control trial with 7M users across 80 countries. Our findings confirm that we can effectively incorporate sequential preferences of users into our sequencer to make users complete more and skip less tracks within their listening sessions.

Personalization of playlists is a common feature in music streaming services, but conventional techniques, such as collaborative filtering, rely on explicit assumptions regarding content quality to learn how to make recommendations. Such assumptions often result in misalignment between offline model objectives and online user satisfaction metrics. In this paper, we present a reinforcement learning framework that solves for such limitations by directly optimizing for user satisfaction metrics via the use of a simulated playlist-generation environment. Using this simulator we develop and train a modified Deep Q-Network, the action head DQN (AH-DQN), in a manner that addresses the challenges imposed by the large state and action space of our RL formulation. The resulting policy is capable of making recommendations from large and dynamic sets of candidate items with the expectation of maximizing consumption metrics. We analyze and evaluate agents offline via simulations that use environment models trained on both public and proprietary streaming datasets. We show how these agents lead to better user-satisfaction metrics compared to baseline methods during online A/B tests. Finally, we demonstrate that performance assessments produced from our simulator are strongly correlated with observed online metric results.

This article presents a novel approach to construct Intrinsic Gaussian Processes for regression on unknown manifolds with probabilistic metrics (GPUM) in point clouds. In many real world applications, one often encounters high dimensional data (e.g.‘point cloud data’) centered around some lower dimensional unknown manifolds. The geometry of manifold is in general different from the usual Euclidean geometry. Naively applying traditional smoothing methods such as Euclidean Gaussian Processes (GPs) to manifold-valued data and so ignoring the geometry of the space can potentially lead to highly misleading predictions and inferences. A manifold embedded in a high dimensional Euclidean space can be well described by a probabilistic mapping function and the corresponding latent space. We investigate the geometrical structure of the unknown manifolds using the Bayesian Gaussian Processes latent variable models(B-GPLVM) and Riemannian geometry. The distribution of the metric tensor is learned using B-GPLVM. The boundary of the resulting manifold is defined based on the uncertainty quantification of the mapping. We use the probabilistic metric tensor to simulate Brownian Motion paths on the unknown manifold. The heat kernel is estimated as the transition density of Brownian Motion and used as the covariance functions of GPUM. The applications of GPUM are illustrated in the simulation studies on the Swiss roll, high dimensional real datasets of WiFi signals and image data examples. Its performance is compared with the Graph Laplacian GP, Graph Matérn GP and Euclidean GP.

Dynamic topic modeling is a well established tool for capturing the temporal dynamics of the topics of a corpus. In this work, we develop a scalable dynamic topic model by utilizing the correlation among the words in the vocabulary. By correlating previously independent temporal processes for words, our new model allows us to reliably estimate the topic representations containing less frequent words. We develop an amortised variational inference method with self-normalised importance sampling approximation to the word distribution that dramatically reduces the computational complexity and the number of variational parameters in order to handle large vocabularies. With extensive experiments on text datasets, we show that our method significantly outperforms the previous works by modeling word correlations, and it is able to handle real world data with a large vocabulary which could not be processed by previous continuous dynamic topic models.

We present a novel approach to Bayesian inference and general Bayesian computation that is defined through a sequential decision loop. Our method defines a recursive partitioning of the sample space. It neither relies on gradients nor requires any problem-specific tuning, and is asymptotically exact for any density function with a bounded domain. The output is an approximation to the whole density function including the normalisation constant, via partitions organised in efficient data structures. Such approximations may be used for evidence estimation or fast posterior sampling, but also as building blocks to treat a larger class of estimation problems. The algorithm shows competitive performance to recent state-of-the-art methods on synthetic and real-world problems including parameter inference for gravitational-wave physics.

Bayesian optimization (BO) methods often rely on the assumption that the objective function is well-behaved, but in practice, this is seldom true for real-world objectives even if noise-free observations can be collected. Common approaches, which try to model the objective as precisely as possible, often fail to make progress by spending too many evaluations modeling irrelevant details. We address this issue by proposing surrogate models that focus on the well-behaved structure in the objective function, which is informative for search, while ignoring detrimental structure that is challenging to model from few observations. First, we demonstrate that surrogate models with appropriate noise distributions can absorb challenging structures in the objective function by treating them as irreducible uncertainty. Secondly, we show that a latent Gaussian process is an excellent surrogate for this purpose, comparing with Gaussian processes with standard noise distributions. We perform numerous experiments on a range of BO benchmarks and find that our approach improves reliability and performance when faced with challenging objective functions.

A challenge that machine learning practitioners in the industry face is the task of selecting the best model to deploy in production. As a model is often an intermediate component of a production system, online controlled experiments such as A/B tests yield the most reliable estimation of the effectiveness of the whole system, but can only compare two or a few models due to budget constraints. We propose an automated online experimentation mechanism that can efficiently perform model selection from a large pool of models with a small number of online experiments. We derive the probability distribution of the metric of interest that contains the model uncertainty from our Bayesian surrogate model trained using historical logs. Our method efficiently identifies the best model by sequentially selecting and deploying a list of models from the candidate set that balance exploration-exploitation. Using simulations based on real data, we demonstrate the effectiveness of our method on two different tasks.

Correlated topic models (CTM) are useful tools for statistical analysis of documents. They explicitly capture the correlation between topics associated with each document. We propose an extension to CTM that models the evolution of both topic correlation and word co-occurrence over time. This allows us to identify the changes of topic correlations over time, e.g., in the machine learning literature, the correlation between the topics “stochastic gradient descent” and “variational inference” increased in the last few years due to advances in stochastic variational inference methods. Our temporal dynamic priors are based on Gaussian processes (GPs), allowing us to capture diverse temporal behaviours such as smooth, with long-term memory, temporally concentrated, and periodic. The evolution of topic correlations is modeled through generalised Wishart processes (GWPs). We develop a stochastic variational inference method, which enables us to handle large sets of continuous temporal data. Our experiments applied to real world data demonstrate that our model can be used to effectively discover temporal patterns of topic distributions, words associated to topics and topic relationships.

In modern plants, industrial processes typically operate under different states to meet the different requirements of high-quality products. Many monitoring models for industrial processes were constructed based on the prior knowledge (the mechanism's model or the process data characteristics) to monitor such processes (called multimode processes). However, obtaining this prior knowledge is difficult in practice. Efficiently monitoring nonlinear multimode processes without any prior knowledge is an open problem that demands further exploration. Since data from different modes follow different distributions while data from the same mode are considered to be sampled from the same distribution, the modes of multimode processes can be uncovered based on the characteristics of the process data. This article proposes using a Dirichlet process Gaussian mixed model to classify the modes of multimode processes based on historical data, and then, determine the mode types of the monitored data. A nonlinear monitoring strategy based on the t-distributed stochastic neighbor embedding is then proposed to achieve nonlinear dimensionality reduction and visualize the data. Finally, a monitoring index that is integrated with support vector data description is constructed for comprehensive monitoring. The proposed nonlinear multimode framework completely realizes data-driven mode identification and unsupervised fault detection without knowing any prior knowledge. The effectiveness and feasibility of the proposed model are demonstrated using data from a simulated wastewater treatment plant.

Transferring knowledge from a teacher neural network pretrained on the same or a similar task to a student neural network can significantly improve the performance of the student neural network. Existing knowledge transfer approaches match the activations or the corresponding hand-crafted features of the teacher and the student networks. We propose an information-theoretic framework for knowledge transfer which formulates knowledge transfer as maximizing the mutual information between the teacher and the student networks. We compare our method with existing knowledge transfer methods on both knowledge distillation and transfer learning tasks and show that our method consistently outperforms existing methods. We further demonstrate the strength of our method on knowledge transfer across heterogeneous network architectures by transferring knowledge from a convolutional neural network (CNN) to a multi-layer perceptron (MLP) on CIFAR-10. The resulting MLP significantly outperforms the-state-of-the-art methods and it achieves similar performance to the CNN with a single convolutional layer.

Despite the recent progress in hyperparameter optimization (HPO), available benchmarks that resemble real-world scenarios consist of a few and very large problem instances that are expensive to solve. This blocks researchers and practitioners no only from systematically running large-scale comparisons that are needed to draw statistically significant results but also from reproducing experiments that were conducted before. This work proposes a method to alleviate these issues by means of a meta-surrogate model for HPO tasks trained on off-line generated data. The model combines a probabilistic encoder with a multi-task model such that it can generate inexpensive and realistic tasks of the class of problems of interest. We demonstrate that benchmarking HPO methods on samples of the generative model allows us to draw more coherent and statistically significant conclusions that can be reached orders of magnitude faster than using the original tasks. We provide evidence of our findings for various HPO methods on a wide class of problems.

We investigate how the neural processing in auditory cortex is shaped by the statistics of natural sounds. Hypothesising that auditory cortex (A1) represents the structural primitives out of which sounds are composed, we employ a statistical model to extract such components. The input to the model are cochleagrams which approximate the non-linear transformations a sound undergoes from the outer ear, through the cochlea to the auditory nerve. Cochleagram components do not superimpose linearly, but rather according to a rule which can be approximated using the max function. This is a consequence of the compression inherent in the cochleagram and the sparsity of natural sounds. Furthermore, cochleagrams do not have negative values. Cochleagrams are therefore not matched well by the assumptions of standard linear approaches such as sparse coding or ICA. We therefore consider a new encoding approach for natural sounds, which combines a model of early auditory processing with maximal causes analysis (MCA), a sparse coding model which captures both the non-linear combination rule and non-negativity of the data. An efficient truncated EM algorithm is used to fit the MCA model to cochleagram data. We characterize the generative fields (GFs) inferred by MCA with respect to in vivo neural responses in A1 by applying reverse correlation to estimate spectro-temporal receptive fields (STRFs) implied by the learned GFs. Despite the GFs being non-negative, the STRF estimates are found to contain both positive and negative subfields, where the negative subfields can be attributed to explaining away effects as captured by the applied inference method. A direct comparison with ferret A1 shows many similar forms, and the spectral and temporal modulation tuning of both ferret and model STRFs show similar ranges over the population. In summary, our model represents an alternative to linear approaches for biological auditory encoding while it captures salient data properties and links inhibitory subfields to explaining away effects.

We tackle the problem of optimizing a black-box objective function defined over a highly-structured input space. This problem is ubiquitous in science and engineering. In machine learning, inferring the structure of a neural network or the Automatic Statistician (AS), where the optimal kernel combination for a Gaussian process is selected, are two important examples. We use the \as as a case study to describe our approach, that can be easily generalized to other domains. We propose an Structure Generating Variational Auto-encoder (SG-VAE) to embed the original space of kernel combinations into some low-dimensional continuous manifold where Bayesian optimization (BO) ideas are used. This is possible when structural knowledge of the problem is available, which can be given via a simulator or any other form of generating potentially good solutions. The right exploration-exploitation balance is imposed by propagating into the search the uncertainty of the latent space of the SG-VAE, that is computed using variational inference. The key aspect of our approach is that the SG-VAE can be used to bias the search towards relevant regions, making it suitable for transfer learning tasks. Several experiments in various application domains are used to illustrate the utility and generality of the approach described in this work.

Bayesian optimization (BO) has emerged during the last few years as an effective approach to optimize black-box functions where direct queries of the objective are expensive. We consider the case where direct access to the function is not possible, but information about user preferences is. Such scenarios arise in problems where human preferences are modeled, such as A/B tests or recommender systems. We present a new framework for this scenario that we call Preferential Bayesian Optimization (PBO) and that allows to find the optimum of a latent function that can only be queried through pairwise comparisons, so-called duels. PBO extend the applicability of standard BO ideas and generalizes previous discrete dueling approaches by modeling the probability of the the winner of each duel by means of Gaussian process model with a Bernoulli likelihood. The latent preference function is used to define a family of acquisition functions that extend usual policies used in BO. We illustrate the benefits of PBO in a variety of experiments in which we show how the way correlations are modeled is the key ingredient to drastically reduce the number of comparisons to find the optimum of the latent function of interest.

Often in machine learning, data are collected as a combination of multiple conditions, e.g., the voice recordings of multiple persons, each labeled with an ID. How could we build a model that captures the latent information related to these conditions and generalize to a new one with few data? We present a new model called Latent Variable Multiple Output Gaussian Processes (LVMOGP) that allows to jointly model multiple conditions for regression and generalize to a new condition with a few data points at test time. LVMOGP infers the posteriors of Gaussian processes together with a latent space representing the information about different conditions. We derive an efficient variational inference method for LVMOGP for which the computational complexity is as low as sparse Gaussian processes. We show that LVMOGP significantly outperforms related Gaussian process methods on various tasks with both synthetic and real data.

MOTIVATION: Regulation of gene expression in prokaryotes involves complex co-regulatory mechanisms involving large numbers of transcriptional regulatory proteins and their target genes. Uncovering these genome-scale interactions constitutes a major bottleneck in systems biology. Sparse latent factor models, assuming activity of transcription factors (TFs) as unobserved, provide a biologically interpretable modelling framework, integrating gene expression and genome-wide binding data, but at the same time pose a hard computational inference problem. Existing probabilistic inference methods for such models rely on subjective filtering and suffer from scalability issues, thus are not well-suited for realistic genome-scale applications. RESULTS: We present a fast Bayesian sparse factor model, which takes input gene expression and binding sites data, either from ChIP-seq experiments or motif predictions, and outputs active TF-gene links as well as latent TF activities. Our method employs an efficient variational Bayes scheme for model inference enabling its application to large datasets which was not feasible with existing MCMC-based inference methods for such models. We validate our method on synthetic data against a similar model in the literature, employing MCMC for inference, and obtain comparable results with a small fraction of the computational time. We also apply our method to large-scale data from Mycobacterium tuberculosis involving ChIP-seq data on 113 TFs and matched gene expression data for 3863 putative target genes. We evaluate our predictions using an independent transcriptomics experiment involving over-expression of TFs. AVAILABILITY AND IMPLEMENTATION: An easy-to-use Jupyter notebook demo of our method with data is available at https://github.com/zhenwendai/SITAR. CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

We propose a nonparametric procedure to achieve fast inference in generative graphical models when the number of latent states is very large. The approach is based on iterative latent variable preselection, where we alternate between learning a selection function to reveal the relevant latent variables and using this to obtain a compact approximation of the posterior distribution for EM. This can make inference possible where the number of possible latent states is, for example, exponential in the number of latent variables, whereas an exact approach would be computationally infeasible. We learn the selection function entirely from the observed data and current expectation-maximization state via gaussian process regression. This is in contrast to earlier approaches, where selection functions were manually designed for each problem setting. We show that our approach performs as well as these bespoke selection functions on a wide variety of inference problems. In particular, for the challenging case of a hierarchical model for object localization with occlusion, we achieve results that match a customized state-of-the-art selection method at a far lower computational cost.

The popularity of Bayesian optimization methods for efficient exploration of parameter spaces has lead to a series of papers applying Gaussian processes as surrogates in the optimization of functions. However, most proposed approaches only allow the exploration of the parameter space to occur sequentially. Often, it is desirable to simultaneously propose batches of parameter values to explore. This is particularly the case when large parallel processing facilities are available. These could either be computational or physical facets of the process being optimized. Batch methods, however, require the modeling of the interaction between the different evaluations in the batch, which can be expensive in complex scenarios. We investigate this issue and propose a highly effective heuristic based on an estimate of the function’s Lipschitz constant that captures the most important aspect of this interaction–local repulsion–at negligible computational overhead. A penalized acquisition function is used to collect batches of points minimizing the non-parallelizable computational effort. The resulting algorithm compares very well, in run-time, with much more elaborate alternatives.

This article deals with the detection of prominent objects in images. As opposed to the standard approaches based on sliding windows, we study a fundamentally different solution by formulating the supervised prediction of a bounding box as an image retrieval task. Indeed, given a global image descriptor, we find the most similar images in an annotated dataset, and transfer the object bounding boxes. We refer to this approach as data-driven detection (DDD). Our key novelty is to design or learn image similarities that explicitly optimize some aspect of the transfer unlike previous work which uses generic representations and unsupervised similarities. In a first variant, we explicitly learn to transfer, by adapting a metric learning approach to work with image and bounding box pairs. Second, we use a representation of images as object probability maps computed from low-level patch classifiers. Experiments show that these two contributions yield in some cases comparable or better results than standard sliding window detectors - despite its conceptual simplicity and run-time efficiency. Our third contribution is an application of prominent object detection, where we improve fine-grained categorization by pre-cropping images with the proposed approach. Finally, we also extend the proposed approach to detect multiple parts of rigid objects.

We study the task of cleaning scanned text documents that are strongly corrupted by dirt such as manual line strokes, spilled ink, etc. We aim at autonomously removing such corruptions from a single letter-size page based only on the information the page contains. Our approach first learns character representations from document patches without supervision. For learning, we use a probabilistic generative model parameterizing pattern features, their planar arrangements and their variances. The model's latent variables describe pattern position and class, and feature occurrences. Model parameters are efficiently inferred using a truncated variational EM approach. Based on the learned representation, a clean document can be recovered by identifying, for each patch, pattern class and position while a quality measure allows for discrimination between character and non-character patterns. For a full Latin alphabet we found that a single page does not contain sufficiently many character examples. However, even if heavily corrupted by dirt, we show that a page containing a lower number of character types can efficiently and autonomously be cleaned solely based on the structural regularity of the characters it contains. In different example applications with different alphabets, we demonstrate and discuss the effectiveness, efficiency and generality of the approach.

We study optimal image encoding based on a generative approach with non-linear feature combinations and explicit position encoding. By far most approaches to unsupervised learning learning of visual features, such as sparse coding or ICA, account for translations by representing the same features at different positions. Some earlier models used a separate encoding of features and their positions to facilitate invariant data encoding and recognition. All probabilistic generative models with explicit position encoding have so far assumed a linear superposition of components to encode image patches. Here, we for the first time apply a model with non-linear feature superposition and explicit position encoding. By avoiding linear superpositions, the studied model represents a closer match to component occlusions which are ubiquitous in natural images. In order to account for occlusions, the non-linear model encodes patches qualitatively very different from linear models by using component representations separated into mask and feature parameters. We first investigated encodings learned by the model using artificial data with mutually occluding components. We find that the model extracts the components, and that it can correctly identify the occlusive components with the hidden variables of the model. On natural image patches, the model learns component masks and features for typical image components. By using reverse correlation, we estimate the receptive fields associated with the model's hidden units. We find many Gabor-like or globular receptive fields as well as fields sensitive to more complex structures. Our results show that probabilistic models that capture occlusions and invariances can be trained efficiently on image patches, and that the resulting encoding represents an alternative model for the neural encoding of images in the primary visual cortex.

We study unsupervised learning of occluding objects in images of visual scenes. The derived learning algorithm is based on a probabilistic generative model which parameterizes object shapes, object features and the background. No assumptions are made for the object orders in depth or the objects' planar positions. Parameter optimization is thus subject to the large combinatorics of depth orders and positions. Previous approaches constrained this combinatorics but were still only able to learn a very small number of objects. By applying a novel variational EM approach, we show that even without constraints on the object combinatorics, a relatively large number of objects can be learned. In different numerical experiments, our unsupervised approach extracts explicit object representations with object masks and object features closely aligned with the true objects in the scenes. We investigate the robustness of the approach and the use of the learned representations for inference. Furthermore, we demonstrate generality of the approach by applying it to grayscale images, color-vector images, and Gabor-vector images as well as to motion trajectory data for which the extracted components correspond to motion primitives.

We study the task of cleaning scanned text documents that are strongly corrupted by dirt such as manual line strokes, spilled ink etc. We aim at autonomously removing dirt from a single letter-size page based only on the information the page contains. Our approach, therefore, has to learn character representations without supervision and requires a mechanism to distinguish learned representations from irregular patterns. To learn character representations, we use a probabilistic generative model parameterizing pattern features, feature variances, the features' planar arrangements, and pattern frequencies. The latent variables of the model describe pattern class, pattern position, and the presence or absence of individual pattern features. The model parameters are optimized using a novel variational EM approximation. After learning, the parameters represent, independently of their absolute position, planar feature arrangements and their variances. A quality measure defined based on the learned representation then allows for an autonomous discrimination between regular character patterns and the irregular patterns making up the dirt. The irregular patterns can thus be removed to clean the document. For a full Latin alphabet we found that a single page does not contain sufficiently many character examples. However, even if heavily corrupted by dirt, we show that a page containing a lower number of character types can efficiently and autonomously be cleaned solely based on the structural regularity of the characters it contains. In different examples using characters from different alphabets, we demonstrate generality of the approach and discuss its implications for future developments.

This paper addresses the problem of estimating the poses of a reference plane in specular shape recovery. Unlike existing methods which require an extra mirror or an extra reference plane and camera, our proposed method recovers the poses of the reference plane directly from its reflections on the specular surface. By establishing reflection correspondences on the reference plane in three distinct poses, our method estimates the poses of the reference plane in two steps. First, by applying a colinearity constraint to the reflection correspondences, a simple closed-form solution is derived for recovering the poses of the reference plane relative to its initial pose. Second, by applying a ray incidence constraint to the incident rays formed by the reflection correspondences and the visual rays cast from the image, a closed-form solution is derived for recovering the poses of the reference plane relative to the camera. The shape of the specular surface then follows. Experimental results on both synthetic and real data are presented, which demonstrate the feasibility and accuracy of our proposed method.