اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدHigh-Dimensional Inference with the generalized Hopfield Model: Principal Component Analysis and Corrections
114
0
0.0
(
0
)
تأليف
Simona Cocco
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the problem of inferring the interactions between a set of N binary variables from the knowledge of their frequencies and pairwise correlations. The inference framework is based on the Hopfield model, a special case of the Ising model where the interaction matrix is defined through a set of patterns in the variable space, and is of rank much smaller than N. We show that Maximum Lik elihood inference is deeply related to Principal Component Analysis when the amp litude of the pattern components, xi, is negligible compared to N^1/2. Using techniques from statistical mechanics, we calculate the corrections to the patterns to the first order in xi/N^1/2. We stress that it is important to generalize the Hopfield model and include both attractive and repulsive patterns, to correctly infer networks with sparse and strong interactions. We present a simple geometrical criterion to decide how many attractive and repulsive patterns should be considered as a function of the sampling noise. We moreover discuss how many sampled configurations are required for a good inference, as a function of the system size, N and of the amplitude, xi. The inference approach is illustrated on synthetic and biological data.
قيم البحث
اقرأ أيضاً
There are several cutting edge applications needing PCA methods for data on tori and we propose a novel torus-PCA method with important properties that can be generally applied. There are two existing general methods: tangent space PCA and geodesic P
CA. However, unlike tangent space PCA, our torus-PCA honors the cyclic topology of the data space whereas, unlike geodesic PCA, our torus-PCA produces a variety of non-winding, non-dense descriptors. This is achieved by deforming tori into spheres and then using a variant of the recently developed principle nested spheres analysis. This PCA analysis involves a step of small sphere fitting and we provide an improved test to avoid overfitting. However, deforming tori into spheres creates singularities. We introduce a data-adaptive pre-clustering technique to keep the singularities away from the data. For the frequently encountered case that the residual variance around the PCA main component is small, we use a post-mode hunting technique for more fine-grained clustering. Thus in general, there are three successive interrelated key steps of torus-PCA in practice: pre-clustering, deformation, and post-mode hunting. We illustrate our method with two recently studied RNA structure (tori) data sets: one is a small RNA data set which is established as the benchmark for PCA and we validate our method through this data. Another is a large RNA data set (containing the small RNA data set) for which we show that our method provides interpretable principal components as well as giving further insight into its structure.
We present a novel technique for sparse principal component analysis. This method, named Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA), is based on the recently detailed formula for computing normed, squared eigenvector l
oadings of a Hermitian matrix from the eigenvalues of the full matrix and associated sub-matrices. Relative to the state-of-the-art LASSO-based sparse PCA method of Witten, Tibshirani and Hastie, the EESPCA technique offers a two-orders-of-magnitude improvement in computational speed, does not require estimation of tuning parameters, and can more accurately identify true zero principal component loadings across a range of data matrix sizes and covariance structures. Importantly, EESPCA achieves these performance benefits while maintaining a reconstruction error close to that generated by the Witten et al. approach. EESPCA is a practical and effective technique for sparse PCA with particular relevance to computationally demanding problems such as the analysis of large data matrices or statistical techniques like resampling that involve the repeated application of sparse PCA.
Unsupervised learning makes manifest the underlying structure of data without curated training and specific problem definitions. However, the inference of relationships between data points is frustrated by the `curse of dimensionality in high-dimensi
ons. Inspired by replica theory from statistical mechanics, we consider replicas of the system to tune the dimensionality and take the limit as the number of replicas goes to zero. The result is the intensive embedding, which is not only isometric (preserving local distances) but allows global structure to be more transparently visualized. We develop the Intensive Principal Component Analysis (InPCA) and demonstrate clear improvements in visualizations of the Ising model of magnetic spins, a neural network, and the dark energy cold dark matter ({Lambda}CDM) model as applied to the Cosmic Microwave Background.
We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any b
lack-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvement over the naive method of combining PCA with regression. To achieve our results, we first observe that ridge regression can be used to obtain a smooth projection onto the top principal components. We then sharpen this approximation to true projection using a low-degree polynomial approximation to the matrix step function. Step function approximation is a topic of long-term interest in scientific computing. We extend prior theory by constructing polynomials with simple iterative structure and rigorously analyzing their behavior under limited precision.
Dimension reduction for high-dimensional compositional data plays an important role in many fields, where the principal component analysis of the basis covariance matrix is of scientific interest. In practice, however, the basis variables are latent
and rarely observed, and standard techniques of principal component analysis are inadequate for compositional data because of the simplex constraint. To address the challenging problem, we relate the principal subspace of the centered log-ratio compositional covariance to that of the basis covariance, and prove that the latter is approximately identifiable with the diverging dimensionality under some subspace sparsity assumption. The interesting blessing-of-dimensionality phenomenon enables us to propose the principal subspace estimation methods by using the sample centered log-ratio covariance. We also derive nonasymptotic error bounds for the subspace estimators, which exhibits a tradeoff between identification and estimation. Moreover, we develop efficient proximal alternating direction method of multipliers algorithms to solve the nonconvex and nonsmooth optimization problems. Simulation results demonstrate that the proposed methods perform as well as the oracle methods with known basis. Their usefulness is illustrated through an analysis of word usage pattern for statisticians.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
