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GATE DA 2025 Set 1 — Question 60

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NAT+2 / -0MediumPrincipal Component Analysis (PCA)Dimensionality ReductionMachine LearningCovarianceDescriptive StatisticsProbability & Statistics

Machine Learning → Descriptive Statistics → Covariance

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Question

Let D={x(1),…,x(n)}D = \{x^{(1)}, \dots, x^{(n)}\} be a dataset of nn observations where each x(i)∈R100x^{(i)} \in \mathbb{R}^{100}. It is given that ∑i=1nx(i)=0\sum_{i=1}^{n} x^{(i)} = 0. The covariance matrix computed from DD has eigenvalues λi=1002−i,1≤i≤100\lambda_i = 100^{2-i}, 1 \leq i \leq 100. Let u∈R100u \in \mathbb{R}^{100} be the direction of maximum variance with u⊤u=1u^\top u = 1. The value of1n∑i=1n(u⊤x(i))2=______\frac{1}{n} \sum_{i=1}^{n} (u^\top x^{(i)})^2 = \_\_\_\_\_\_(Answer in integer)
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