Quantum annealing for linear spectral unmixing: a proof-of-concept

In the last decade, the issue of endmember variability has received considerable attention, particularly when each pixel is modeled as a linear combination of endmembers or pure materials. As a result, several models and algorithms have been developed for considering the effect of endmember variability in spectral unmixing and possibly include multiple endmembers in the spectral unmixing stage. One of the most popular approach for this purpose is the multiple endmember spectral mixture analysis (MESMA) algorithm. The procedure executed by MESMA can be summarized as follows: (i) First, a standard linear spectral unmixing (LSU) or fully constrained linear spectral unmixing (FCLSU) algorithm is run in an iterative fashion; (ii) Then, we use different endmember combinations, randomly selected from a spectral library, to decompose each mixed pixel; (iii) Finally, the model with the best fit, i.e., with the lowest root mean square error (RMSE) in the reconstruction of the original pixel, is adopted. However, this procedure can be computationally very expensive due to the fact that several endmember combinations need to be tested and several abundance estimation steps need to be conducted, a fact that compromises the use of MESMA in applications under real-time constraints. In this paper we develop (for the first time in the literature) an efficient implementation of MESMA on different platforms using OpenCL, an open standard for parallel programing on heterogeneous systems. Our experiments have been conducted using a simulated data set and the clMAGMA mathematical library. This kind of implementations with the same descriptive language on different architectures are very important in order to actually calibrate the possibility of using heterogeneous platforms for efficient hyperspectral imaging processing in real remote sensing missions.

Linear spectral unmixing is a popular tool in remotely sensed hyperspectral data interpretation. It aims at estimating the fractional abundances of pure spectral signatures (also called as endmembers) in each mixed pixel collected by an imaging spectrometer. In many situations, the identification of the end-member signatures in the original data set may be challenging due to insufficient spatial resolution, mixtures happening at different scales, and unavailability of completely pure spectral signatures in the scene. However, the unmixing problem can also be approached in semisupervised fashion, i.e., by assuming that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In practice, this is a combinatorial problem which calls for efficient linear sparse regression (SR) techniques based on sparsity-inducing regularizers, since the number of endmembers participating in a mixed pixel is usually very small compared with the (ever-growing) dimensionality (and availability) of spectral libraries. Linear SR is an area of very active research, with strong links to compressed sensing, basis pursuit (BP), BP denoising, and matching pursuit. In this paper, we study the linear spectral unmixing problem under the light of recent theoretical results published in those referred to areas. Furthermore, we provide a comparison of several available and new linear SR algorithms, with the ultimate goal of analyzing their potential in solving the spectral unmixing problem by resorting to available spectral libraries. Our experimental results, conducted using both simulated and real hyperspectral data sets collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infrared Imaging Spectrometer and spectral libraries publicly available from the U.S. Geological Survey, indicate the potential of SR techniques in the task of accurately characterizing the mixed pixels using the library spectra. This opens new perspectives for spectral unmixing, since the abundance estimation process no longer depends on the availability of pure spectral signatures in the input data nor on the capacity of a certain endmember extraction algorithm to identify such pure signatures.

Spectral Unmixing is a challenging and absorbing problem. Unmixning allows us to break down a pixel's composition into its material components. Many avenues of spectral unmixing have been attempted with considerable success. One such avenue is to frame the spectral unmixing problem as an Estimation-Measurement problem and avail the use of the well-known Kalman Filter (KF) technique. Two such recent work has been the KF based Linear Unmixing (KFLU) approach and the KF approach for Hyperspectral Signature Estimation, Identification and Abundance Quantification (KFHSE/I/AQ). The above techniques aim to address the spectral unmixing and the spectral signature identification problems respectively. This work extends the above formulation by the use of the Particle Filter (PF) based filtering approach. The particle filter is a recent development in the KF framework. It addresses two major improvements over the KF. It enables use of nonlinearity in the estimation process and further allows fusion of multiple information sources. Additionally, by the use of distributed setup using particles, measurement errors are more efficiently reduced. The above enhancements are the primary motivation to create the proposed Kalman Particle Filter (KPF) in this paper. A major disadvantage in the use of Kalman Filter is the selection of the estimation matrix which can be efficiently resolved using the Kalman Particle Filter as described in this paper. Experiments performed using this new algorithm demonstrate the utility of the proposed approach as a better new tool to solve the spectral unmixing problem as compared to prior Kalman linear unmixing approach.

Linear spectral unmixing is the practice of decomposing the mixed pixel into a linear combination of the constituent endmembers and the estimated abundances. This paper focuses on unsupervised spectral unmixing where the endmembers are unknown a priori . Conventional approaches use either geometrical- or statistical-based approaches. In this paper, we address the challenges of spectral unmixing with unsupervised deep learning models, in specific, the autoencoder models, where the decoder serves as the endmembers and the hidden layer output serves as the abundances. In several recent attempts, part-based autoencoders have been designed to solve the unsupervised spectral unmixing problem. However, the performance has not been satisfactory. In this paper, we first discuss some important findings we make on issues with part-based autoencoders. By proof of counterexample, we show that all existing part-based autoencoder networks with nonnegative and tied encoder and decoder are inherently defective by making these inappropriate assumptions on the network structure. As a result, they are not suitable for solving the spectral unmixing problem. We propose a so-called untied denoising autoencoder with sparsity, in which the encoder and decoder of the network are independent, and only the decoder of the network is enforced to be nonnegative. Furthermore, we make two critical additions to the network design. First, since denoising is an essential step for spectral unmixing, we propose to incorporate the denoising capacity into the network optimization in the format of a denoising constraint rather than cascading another denoising preprocessor in order to avoid the introduction of additional reconstruction error. Second, to be more robust to the inaccurate estimation of a number of endmembers, we adopt an $l_{21}$ -norm on the encoder of the network to reduce the redundant endmembers while decreasing the reconstruction error simultaneously. The experimental results demonstrate that the proposed approach outperforms several state-of-the-art methods, especially for highly noisy data.

Spectral mixture analysis of AVIRIS-NG hyperspectral data for material identification and classification for the part of Kolkata city

Spectral unmixing is an important data processing task that is commonly applied to hyperspectral imaging data. It uses a set of spectral pixels, i.e., multichannel data, to separate each spectral pixel, a multicomponent signal comprised of a linear mixture of pure spectral signatures, into its individual spectral signatures commonly known as endmembers. When no prior information about the required endmembers is available, the resulting unsupervised unmixing problem will be underdetermined, and additional constraints become necessary. A recent approach to solving this problem required that these endmembers be sparse in some dictionary. Sparse signal recovery is commonly solved using a basis pursuit optimization algorithm that requires specifying a data-dependent regularization parameter. Least angle regression (LARS) is a very efficient method to simultaneously solve the basis pursuit optimization problem for all relevant regularization parameter values. However, despite this efficiency of LARS, it has not been applied to the spectral unmixing problem before. This is likely because the application of LARS to large multichannel data could be very challenging in practice, due to the need for generation and storage of extremely large arrays (~ 1010 bytes in a relatively small area of spectral unmixing problem). In this paper, we extend the standard LARS algorithm, using Kronecker products, to make it suitable for practical and efficient recovery of sparse signals from large multichannel data, i.e., without the need to construct or process very large arrays, or the need for trial and error to determine the regularization parameter value. We then apply this new Kronecker LARS (K-LARS) algorithm to successfully achieve spectral unmixing of both synthetic and AVIRIS hyperspectral imaging data. We also compare our results to ones obtained using an earlier basis pursuit-based spectral unmixing algorithm, generalized morphological component analysis (GMCA). We show that these two results are similar, albeit our results were obtained without trial and error, or arbitrary choices, in specifying the regularization parameter. More important, our K-LARS algorithm could be a very valuable research tool to the signal processing community, where it could be used to solve sparse least squares problems involving large multichannel data.

The observation of an unequivocal quantum speedup remains an elusive objective for quantum computing. The D-Wave quantum annealing processors have been at the forefront of experimental attempts to address this goal, given their relatively large numbers of qubits and programmability. A complete determination of the optimal time-to-solution (TTS) using these processors has not been possible to date, preventing definitive conclusions about the presence of a scaling advantage. The main technical obstacle has been the inability to verify an optimal annealing time within the available range. Here we overcome this obstacle and present a class of problem instances for which we observe an optimal annealing time using a D-Wave 2000Q processor over a range spanning up to more than $2000$ qubits. This allows us to perform an optimal TTS benchmarking analysis and perform a comparison to several classical algorithms, including simulated annealing, spin-vector Monte Carlo, and discrete-time simulated quantum annealing. We establish the first example of a scaling advantage for an experimental quantum annealer over classical simulated annealing: we find that the D-Wave device exhibits certifiably better scaling than simulated annealing, with $95\%$ confidence, over the range of problem sizes that we can test. However, we do not find evidence for a quantum speedup: simulated quantum annealing exhibits the best scaling by a significant margin. Our construction of instance classes with verifiably optimal annealing times opens up the possibility of generating many new such classes, paving the way for further definitive assessments of scaling advantages using current and future quantum annealing devices.

The Potts model is a generalization of the Ising model with $Q>2$ components. In the fully connected ferromagnetic Potts model, a first-order phase transition is induced by varying thermal fluctuations. Therefore, the computational time required to obtain the ground states by simulated annealing exponentially increases with the system size. This study analytically confirms that the transverse magnetic-field quantum annealing induces a first-order phase transition. This result implies that quantum annealing does not exponentially accelerate the ground-state search of the ferromagnetic Potts model. To avoid the first-order phase transition, we propose an iterative optimization method using a half-hot constraint that is applicable to both quantum and simulated annealing. In the limit of $Q \to \infty$, a saddle point equation under the half-hot constraint is identical to the equation describing the behavior of the fully connected ferromagnetic Ising model, thus confirming a second-order phase transition. Furthermore, we verify the same relation between the fully connected Potts glass model and the Sherrington--Kirkpatrick model under assumptions of static approximation and replica symmetric solution. The proposed method is expected to obtain low-energy states of the Potts models with high efficiency using Ising-type computers such as the D-Wave quantum annealer and the Fujitsu Digital Annealer.

In recent years, the diversification of problems that require computers to solve has attracted attention to the construction of meta-heuristics that can be applied to a wide range of problems, and to specialized computers that implement these meta-heuristics in their devices. The representative meta-heuristics are Simulated Annealing (SA) and its extension to quantum computation, Quantum Annealing (QA), and its path-integral Monte Carlo method for classical simulation Crosson and Harrow showed that for certain problems where QA outperformed SA, SQA achieved performance close to that of QA, and SQA sometimes outperformed SA by an exponential time factor. On the other hand, it remains unclear whether SQA can work efficiently on a wide range of other problems. In this study, we experimentally compared SA and SQA on instances of the restricted Boltzmann machine RBM, known as a fundamental building block in deep learning, and 3SAT, a fundamental combinatorial optimization problem. The results show that SQA gives slightly better solutions than SA as the problem size increases for RBM in terms of both accuracy and computation time in our setting, but the opposite trend is observed for 3SAT, indicating that there is no significant difference between the two methods. From the viewpoint of artificial intelligence research, it is necessary to further examine whether deep learning can be made more efficient by applying QA and SQA to RBM.

Quantum annealing (QA) serves as a specialized optimizer that is able to solve many NP-hard problems and that is believed to have a theoretical advantage over simulated annealing (SA) via quantum tunneling. With the introduction of the D-Wave programmable quantum annealer, a considerable amount of effort has been devoted to detect and quantify quantum speedup. While the debate over speedup remains inconclusive as of now, instead of attempting to show general quantum advantage, here, we focus on a novel real-world application of D-Wave in wireless networking—more specifically, the scheduling of the activation of the air-links for maximum throughput subject to interference avoidance near network nodes. In addition, D-Wave implementation is made error insensitive by a novel Hamiltonian extra penalty weight adjustment that enlarges the gap and substantially reduces the occurrence of interference violations resulting from inevitable spin bias and coupling errors. The major result of this paper is that quantum annealing benefits more than simulated annealing from this gap expansion process, both in terms of ST99 speedup and network queue occupancy. It is the hope that this could become a real-word application niche where potential benefits of quantum annealing could be objectively assessed.

Quantum annealing is a promising tool for solving optimization problems, similar in some ways to the traditional (classical) simulated annealing of Kirkpatrick et al. Simulated annealing takes advantage of thermal fluctuations in order to explore the optimization landscape of the problem at hand, whereas quantum annealing employs quantum fluctuations. Intriguingly, quantum annealing has been proved to be more effective than its classical counterpart in many applications. We illustrate the theory and the practical implementation of both classical and quantum annealing – highlighting the crucial differences between these two methods – by means of results recently obtained in experiments, in simple toy-models, and more challenging combinatorial optimization problems (namely, Random Ising model and Travelling Salesman Problem). The techniques used to implement quantum and classical annealing are either deterministic evolutions, for the simplest models, or Monte Carlo approaches, for harder optimization tasks. We discuss the pro and cons of these approaches and their possible connections to the landscape of the problem addressed.

An endmember optimization approach for linear spectral unmixing of fine-scale urban imagery

Mapping urban surfaces using Earth Observation data is one the most challenging tasks of remote sensing field, because of the high spatial and spectral diversity of man-made structures. Spectral unmixing techniques, although designed and mainly used with hyperspectral data, can be proven useful for use with spectral data as well to assess sub-pixel information. For urban areas, the large spectral variability imposes the use of multiple endmember spectral mixture analysis techniques, which are very demanding in terms of computation time. In this study, an artificial neural network is used to inverse the pixel spectral mixture in Landsat imagery. To train the network, a spectal library was created, consisting of pure endmember spectra collected from the image and synthetic mixed spectra produced from combinations of the pure ones. Among the advantages of using a neural network is its low computational demand and its ability to capture non-linearities in the spectral mixture.

Monitoring two decades of urbanization in the Poyang Lake area, China through spectral unmixing

The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power-law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an existing phenomenological theory for this problem. Performance of simulated annealing is shown to be enhanced by introducing quasiglobal searches across energy barriers, leading to a power-law convergence but with a smaller power than in the quantum case and thus a slower convergence classically even with quasiglobal search processes. We also reveal how diabatic quantum dynamics, quantum tunneling in particular, steers the systems toward the global minimum by a meticulous choice of annealing schedule. This latter result explicitly contrasts the role of tunneling in quantum annealing against the classical counterpart of stochastic optimization by simulated annealing.