Compressive Sensing

Junwei Lu
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Junwei Lu: Harvard University

Chapter Chapter 8 in Big Data Analysis, 2025, pp 47-52 from Springer

Abstract: Abstract In the high-dimensional setting, we’re essentially looking at the same linear model Y = 𝕏 β + ε $$Y = \mathbb {X} \beta + \varepsilon $$ with 𝕏 ∈ ℝ n × d $$\mathbb {X} \in \mathbb {R}^{n \times d}$$ . However, we now expect the number of features d is much larger than its sample size n. Under the high-dimensional setting, the ordinary least squares estimator will have troubles. If the features are linearly independent, we have rank ( 𝕏 ) = n $$\mathrm {rank} (\mathbb {X}) = n$$ . Then, 𝕏 β ^ LS = P 𝕏 Y = Y $$\mathbb {X} \widehat {\beta}^{\mathrm {LS}} = P_{\mathbb {X}} Y = Y$$ , i.e., the ordinary least squares will overfit. Therefore, we need to invoke the parsimonious principle and introduce the following sparse linear model.

Date: 2025
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DOI: 10.1007/978-3-032-03161-7_8

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