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What are the limitations of ICA when the data dimension is much larger than the sample size? 


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ICA faces limitations in high dimensions compared to sample size due to instability, especially when independent components vary with each run . In the high-dimensional scaling limit, the joint empirical measure of the target feature vector and algorithm estimates converges to a deterministic process, aiding performance evaluation . High dimension, low sample size data analysis lacks attention to sample integrity, potentially leading to outliers . Methodologies like Dynamic Tilted Current Correlation Screening (DTCCS) address challenges in high-dimensional statistics, offering variable screening and influence measure techniques . Fuzzy cluster scaled PCA is proposed for dimensional reduction in high dimension, low sample size data, enabling discrimination of individual subjects observed by body-worn sensors .

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ICA faces limitations with high dimension low sample size data due to incorrect solutions from the covariance matrix eigenvalues, hindering multivariate analyses, as discussed in the paper.
ICA faces challenges in 'large p, small n' settings due to spurious correlations and inadequate predictor relevance detection, necessitating new methodologies like DTCCS, HD-HIM, and Cosine PoSI for high-dimensional data analysis.
ICA faces challenges in high dimension, low sample size data due to sample integrity issues like outlier detection, which have not received sufficient attention in analysis.
ICA in high dimensions faces instability due to algorithm stochasticity and noise. Minimum Distance index limitations arise, becoming concentrated and suboptimal. Visual inspection remains crucial for effective denoising in high dimensions.
Open accessPosted Content
Chuang Wang, Yue Lu 
15 Oct 2017-arXiv: Learning
9 Citations
In the high-dimensional scaling limit, ICA's performance is analyzed, showing convergence to a deterministic process via a nonlinear PDE, offering insights for efficient algorithm design.

Related Questions

What are the limitations of PCA when the data dimension is much larger than the sample size?5 answersPCA faces limitations when the data dimension is significantly larger than the sample size. In such scenarios, PCA can exhibit unexpected behaviors like upward bias in sample eigenvalues and inconsistency in sample eigenvectors. This phenomenon has been extensively studied under the spiked covariance model, highlighting issues such as the behavior of sample eigenvalues and the rescaled extreme sample eigenvalues. To address these challenges, alternative estimation methods have been developed, leveraging sparsity in eigenvectors or the covariance matrix. Additionally, in high-dimensional low sample size data, obtaining correct solutions through traditional multivariate analyses becomes mathematically challenging, necessitating innovative approaches like fuzzy cluster scaled PCA for effective dimensional reduction.
What methods of analysis are available for high-dimensional small sample data?5 answersMethods of analysis for high-dimensional small sample data include the pairwise Hotelling method, bias-adjustment methods, feature selection and extraction methods, max t-test-type statistics, and dynamic eigenvector estimation. The pairwise Hotelling method balances existing approaches by utilizing correlation information and provides good control over type I error rates and statistical power, especially for highly correlated covariates. Bias-adjustment methods adjust for bias in principal component scores and improve classification performance in high-dimensional situations with small sample sizes. Feature selection and extraction methods, such as Correlation Explanation (CorEx), Conditional Infomax Feature Extraction (CIFE), Maximum Relevance Minimum Redundancy (MRMR), and Kernel Entropy Component Analysis (KECA), can be used to classify high-dimensional low sample size datasets. Max t-test-type statistics, analyzed using a randomization-based approach, are suitable for analyzing high-dimensional designs with small sample sizes. Dynamic eigenvector estimation allows for modeling the change in high-dimensional data over time and can be effectively solved using manifold optimization algorithms.
What are the limitations of AI?5 answersAI has several limitations. Content-filtering AI systems used by online service providers and governments to regulate internet content are subject to limitations that affect their accuracy and transparency, potentially leading to the removal of legitimate content and the persistence of objectionable content. Implementing AI in administrative procedures requires careful consideration of the capabilities and limitations of different systems, including transparency and data availability. The validity of AI predictions relies on extensive, representative, and unbiased training data, and transparency is crucial for establishing trust and reliability in AI systems. Additionally, AI systems in healthcare, such as those used for colorectal cancer, rely on reductive reasoning and computational determinism, which can lead to inherent biases, limited evaluations, and the marginalization of socio-technical factors, posing risks to clinical decision-making and patient care. Furthermore, the proliferation of AI systems is currently benefiting those with economic power, potentially exacerbating global inequality.
Is bootstrapping ever used when performing PCA on data where the sample size is less than a dimension?4 answersBootstrapping is used in PCA to overcome the small sample problem and improve the mean centering process. It is shown that when using the bootstrap procedure, a small number of synthesized images can effectively multiply biometric data without degradation of correlations, leading to the possibility of reducing the size of the test biometric sample. Bootstrap methods are rarely used in conjunction with propensity-score matching, but two different bootstrap methods have been proposed for use in this context, with one method involving drawing bootstrap samples from the matched pairs in the propensity-score-matched sample. The bootstrap method can be used to conduct relevant causal tests on big data, such as in the field of stock market analysis.
Is the performance of ICA worse in short time series?5 answersThe performance of ICA on short time series varies depending on the specific application. In the study by Lopez et al., the performance of multifractal detrended analysis using ICA was evaluated on short time series, including the daily exchange rate between the U.S. dollar and the euro. In another study by James et al., the authors analyzed intracranial electroencephalogram recordings using ICA and found that standard ensemble ICA resulted in marginal improvements in separation for short time series, while space-time ICA using just 2 recording channels showed more distinct components. Peña et al. compared the performance of different ICA procedures on multivariate time series data and found that one of the procedures, FOTBI, provided accurate forecasts for the industrial production index time series. Lu et al. proposed an ICA-based signal reconstruction approach for financial time series forecasting and demonstrated its superiority over other models. Finally, Hou et al. used ICA for weighted filtering and extraction of common mode errors in a regional GNSS observation network, showing improved precision compared to traditional ICA.
Is it appropriate to perform a PCA when the sample size is less than the variable?4 answersPerforming PCA when the sample size is less than the number of variables can lead to spurious results and should be done with caution. Sample correlation matrices will always result in decreasing eigenvalues, even if there is no structure, making it necessary to test for real patterns. PC-scores calculated from non-distinct PC's have large standard errors and cannot be used for biological interpretations. However, there are methods that have been developed to address the issue of high dimension low sample size (HDLSS) settings, such as cross data matrix-based methods, sparse PCA, and spherical PCA. These methods have shown asymptotic normality and consistency for estimates of PC directions in HDLSS settings.

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