# Probabilistic Mixture Model-Based Spectral Unmixing
## 2 Related Work  
In the broad landscape of spectral unmixing, several techniques have emerged, including geometrical methods (e.g., N-FINDR), statistical approaches, non-negative matrix factorization (NMF), and sparse regression-based methods [6, 7]. These methods can be broadly categorized into deterministic and probabilistic approaches. Deterministic methods, which are overwhelmingly more common among the state of the art, include approaches such as sparse regression and simplex-based techniques. These deterministic methods provide point estimates of endmembers and abundances. Some deterministic methods, like sparse unmixing with L1 regularization, can provide a limited form of uncertainty quantification through the selection of regularization parameters. Geometric simplex methods, such as N-FINDR [6], rely on the assumption that the spectral signatures of the endmembers are distinct and that the observed mixed pixels lie within a simplex formed by the endmembers. This assumption is related to the underlying principle of manifold learning techniques, which assume that high-dimensional data lie on a lower-dimensional manifold. N-FINDR includes a dimensionality reduction step using orthogonal subspace projection (OSP) techniques such as SVD, PCA, or MNF before searching for the endmembers. However, while N-FINDR uses linear dimensionality reduction and aims to find the endmembers in the reduced space, manifold learning methods are designed to handle non-linear manifolds and aim to uncover the intrinsic low-dimensional structure of the data. Non-probabilistic machine learning approaches have gained popularity in spectral unmixing due to their ability to learn complex representations and relationships from data. NMF, a widely used technique, decomposes the spectral data matrix into non-negative endmember and abundance matrices [8]. NMF-based methods have been extended to incorporate various constraints and regularizations to improve their performance and to handle different types of spectral variability [9]. Sparse unmixing methods, on the other hand, exploit the fact that each pixel in hyperspectral images is often a mixture of only a few endmembers, incorporating sparsity constraints to estimate the abundances [10; 11]. Manifold learning techniques [12], such as locally linear embedding (LLE) [13] and isometric feature mapping (ISOMAP) [14], have been applied to spectral unmixing by assuming that the high-dimensional spectral data lie on a low-dimensional manifold. These methods aim to uncover the intrinsic structure of the data and can handle non-linear spectral variability to some extent. Dictionary learning approaches, which learn a dictionary of endmember spectra directly from the hyperspectral data, have also shown potential in adapting to the specific characteristics of the input data. More recently, deep learning models, such as convolutional neural networks (CNNs) and autoencoders, have been applied to spectral unmixing [15; 16; 17]. For example, [PERSON] et al. [15] propose an autoencoder-based method for blind hyperspectral unmixing. Their method uses a deep autoencoder to extract endmembers and estimate abundances in an unsupervised manner. The architecture is designed to enforce the linear mixing model, with the decoder weights representing the endmembers. While their approach is unsupervised, scalable, and compares favorably to benchmark methods in reconstruction accuracy, it does not model endmember uncertainties. Generally, while non-probabilistic machine learning approaches have shown promise in addressing specific challenges in spectral unmixing, they typically generate point estimates of the endmember spectral signatures and abundances. Thus, this wide array of non-probabilistic approaches offers advantages in interpretability and robustness (with respect to spectral variability) over conventional methods but falls short in handling both epistemic uncertainties in the endmember spectra and fractional abundances, as well as the aleatoric uncertainties arising from noise in the measurements.