Numerical Tours
Computational tours that put the book’s mathematical ideas into practice through detailed implementations.
Explore the Numerical ToursSignal processing · Optimization · Learning
Mathematical Foundations of Data Sciences
A mathematical route from signals and images to inverse problems, optimization, and machine learning. Read the theory, explore the figures, and connect the ideas to computational methods.
Free to read · Full PDF: 17.2 MB
19chapters, from foundations to applications
145interactive figures to explore
277pages in the complete PDF
Contents
Read chapter by chapter
Follow the book in order or go straight to a topic. Each preview opens a visual example in the online chapter. Separate PDFs are available alongside.
19 chapters
Signal and image processing
Chapters 01–07-
Sampling and aliasing
Shannon Sampling Theory
Sampling, aliasing, and the reconstruction of continuous signals.
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Image spectra
Fourier and Convolution
Fourier representations, convolution, filtering, and the discrete transform.
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Prefix codes
Shannon Coding Theory
Entropy, prefix codes, Huffman coding, and block coding.
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Wavelet coefficients
Wavelets
Multiresolution analysis, filter banks, and wavelet representations.
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Adaptive approximation
Linear and Nonlinear Approximation
Coefficient selection, approximation rates, and geometric image models.
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Image compression
Compression
Quantization, source models, and image compression.
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Removing noise
Denoising
Noise models, filtering, and wavelet thresholding.
Inverse problems and sparsity
Chapters 08–12-
Image gradients
Variational Priors and Regularization
Image gradients, diffusion, and total-variation regularization.
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Tomographic projections
Inverse Problems
Observation operators, reconstruction, and ill-posed problems.
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The geometry of sparsity
Sparse Regularization
Sparse models, regularization paths, and iterative reconstruction.
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Recovery certificates
Theory of Sparse Regularization
Dual certificates, recovery conditions, and stability.
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Fewer measurements
Compressed Sensing
Random measurements and reconstruction from incomplete data.
Learning and optimization
Chapters 13–19-
Principal components
Basics of Machine Learning
Principal components, clustering, regression, and classification.
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Gradient descent
Optimization & Machine Learning: Smooth Optimization
Gradient descent, smoothness, convexity, and convergence.
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Stochastic optimization
Optimization & Machine Learning: Advanced Topics
Stochastic, accelerated, and second-order optimization methods.
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One hidden layer
Shallow Learning
Perceptrons, shallow neural networks, and approximation.
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Convolutional networks
Deep Learning
Deep networks, backpropagation, and learned representations.
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Supporting hyperplanes
Convex Analysis
Convex sets and functions, subgradients, and duality.
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Proximity and projection
Nonsmooth Convex Optimization
Proximity operators, splitting methods, and nonsmooth objectives.
No chapters match these filters.
From mathematical ideas
to numerical methods
This book develops the mathematical foundations behind signal and image processing, inverse problems, and machine learning. Linear operators, nonlinear approximation, and convex optimization provide a common language for understanding the methods and turning them into algorithms.
It is written as a companion to the Numerical Tours of Data Sciences. The interactive edition follows the same chapters and lets you change parameters, compare methods, and inspect the figures alongside the mathematics.
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