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Zero correlation does not always imply independence. However, if two random variables have exponentially decaying tails and all their mixed polynomial covariances vanish, then they must be independent. The proof uses moment generating functions to extend polynomial factorization to the full distribution.
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How Pythagoras and positive definiteness geometrically prove the Cauchy–Schwarz inequality. The inequality simply states that projecting a vector onto another cannot increase its length. This geometric perspective reveals that Cauchy–Schwarz represents the cosine of an angle between vectors, connecting it to the law of cosines and Pearson correlation.
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Natural language search for relational schemas with millisecond latency.
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A systematic treatment of linear regression from the perspective of solving linear systems. We characterize when exact solutions exist based on rank conditions and derive the closed-form solution ($X^T X)^{-1} X^T y$. In the regression setting with noise, this formula emerges as both the least-squares minimizer and the maximum likelihood estimator.
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A decoder-free variational pretraining framework for time-series forecasting under feature asymmetry. VITA learns latent representations that bridge the gap between rich training features and limited deployment features, using a transformer encoder with a seasonal prior. Applied to agricultural yield forecasting, VITA achieves R² = 0.726 on extreme weather years, training in <2.5 hours on a single GPU.
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Predicting FIFA World Cup 2022 match outcomes using Bayesian inference on Poisson processes. The model treats each team’s latent strength as evolving stochastically over time, with observations generated via a Poisson likelihood. This design prioritizes posterior interpretability and uncertainty quantification over raw predictive performance.
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We sped-up binary heap operations 55% using ML.
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A simple proof bounding the prime counting function using Chebyshev’s technique.
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A package for seemless integration of Bangla fonts in large LaTeX projects.