Hello there! I'm excited to share my personal space where I explore the fascinating world of statistical computing and randomized algorithms. Over the years, I've been captivated by the interplay between rigorous probability theory and practical algorithm design. This repository is my personal playground—a place where I dive deep into theory, experiment with implementations, and document my learning journey.
Inspired by two influential books—Statistical Computing in C++ and R by Ana Kupresanin and R. L. Eubank, and Probability and Computing by Michael Mitzenmacher and Eli Upfal—I set out to bridge the gap between abstract theory and real-world applications. Here, I work on projects that range from the fundamentals of probability to advanced randomized algorithms, all while keeping a keen eye on both theoretical elegance and practical performance.
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Foundations of Probability:
I begin with the basics—the axioms of probability and the beauty of events. Early explorations include applications like verifying polynomial identities and matrix multiplication through randomized methods. These concepts form the backbone of the work that follows. -
Discrete Random Variables and Expectation:
Diving into discrete distributions such as Bernoulli and Binomial, I explore the power of expectation. Topics like linearity of expectation and classic challenges such as the Coupon Collector’s Problem or analyzing QuickSort's runtime are central to my studies. -
Moments, Deviations, and Concentration Bounds:
Understanding the variability of random processes is key. I tackle variance, Chebyshev's inequality, and then move into the realm of Chernoff and Hoeffding bounds. These concentration bounds are critical when assessing how far off our algorithms might stray from expected behavior. -
Randomized Algorithms and the Probabilistic Method:
One of my favorite areas is using randomness as a strategic tool. I explore techniques such as the randomized Min-Cut algorithm and methods to derandomize processes via conditional expectations. These topics have taught me that sometimes, a bit of randomness can simplify even the most complex problems. -
Hashing, Random Graphs, and Random Walks:
The study of randomness naturally extends into practical applications like hashing and graph theory. I investigate classic models such as the Balls-and-Bins problem and random graphs, and I delve into Markov chains and random walks—powerful tools for modeling dynamic systems. -
Bayesian Inference and Monte Carlo Methods:
Merging probability with statistics, I experiment with Bayesian methods and Monte Carlo simulations, including the Metropolis algorithm and EM for Gaussian mixtures. These techniques are incredibly useful in making sense of data and in solving high-dimensional problems. -
Entropy, Information Theory, and Universal Hash Functions:
I also find great interest in the theory of information. From Shannon entropy and coding theorems to designing robust universal hash functions, these topics illustrate the profound connections between data, randomness, and algorithmic efficiency.
This repository is a reflection of my personal journey into the heart of statistical computing. Here’s why it matters to me:
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A Passion for Learning:
I have always been fascinated by how a few mathematical principles can unlock efficient solutions to complex problems. This repository is my way of deepening that understanding and continuously challenging myself. -
Bridging Theory and Practice:
I believe that true mastery comes from the marriage of theory and hands-on application. By experimenting with both, I aim to see how abstract ideas can be transformed into practical tools that solve real-world problems. -
Sharing My Journey:
I hope that by putting my work out there, I can connect with others who share similar interests—whether you're a student, a researcher, or simply curious about the magic behind randomized algorithms. I believe that knowledge grows when it's shared and discussed. -
Embracing Creativity and Experimentation:
Not everything works perfectly on the first try, and that's okay! This is a space for trial, error, and creative exploration. I encourage you to embrace this mindset too, and to see failures as stepping stones to deeper insights.
This repository is a living project. As I continue to learn and experiment, you'll find new topics, refined implementations, and fresh insights added over time. My goal is to keep pushing the boundaries of what I know and to document the evolution of my understanding in this exciting field.
Thank you for taking the time to explore my work. I’m thrilled to have you along on this journey of discovery, and I look forward to seeing where these explorations take us both.
Happy exploring!
— Haniel Ulises :D