# inner product Published articles for inner product. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Carnival of Mathematics #197 DevFeed: [Carnival of Mathematics #197]() Original publisher: [Read original article]() Published: 2021-09-01T08:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Graphics](), [Simulation](), [Data Science]() Tags: [carnival](), [dataset](), [fibonacci](), [folding](), [functional-analysis](), [geometric-series](), [graphics](), [inner-product](), [knot-theory](), [mathematics](), [oeis](), [sequence](), [sieve](), [simulation](), [umap](), [visualization]() ### AI overview The 197th Carnival of Mathematics introduces the number 197 and highlights mathematical and technical topics including curve untangling, folding equilateral triangles without measurement, and criticism of UMAP and t-SNE dimensionality reduction. ### Source excerpt Welcome to the 197th Carnival of Mathematics! 197 is an unseemly number, as you can tell by the Wikipedia page which currently says that it has "indiscriminate, excessive, or irrelevant examples." How deviant. It's also a Repfigit, which means if you start a fibonacci-type sequence with the digits 1, 9, 7, and then continue with $ a_n = a_{i-3} + a_{i-2} + a_{i-1}$, then 197 shows up in the sequence. Indeed: 1, 9, 7, 17, 33, 57, 107, 197, ... ## Formulating the Support Vector Machine Optimization Problem DevFeed: [Formulating the Support Vector Machine Optimization Problem]() Original publisher: [Read original article]() Published: 2017-06-05T08:00:46Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Optimization](), [math](), [Machine Learning & Artificial Intelligence](), [data]() Tags: [blog-post](), [data-visualization](), [inner-product](), [javascript](), [machine-learning](), [math](), [mathematics](), [optimization](), [programming](), [support-vector-machines]() ### AI overview A tutorial formulates the support vector machine optimization problem using separating hyperplanes, labeled training data, vector-and-scalar bias terms, and the goal of choosing a separator that generalizes well. ### Source excerpt The hypothesis and the setup This blog post has an interactive demo (mostly used toward the end of the post). The source for this demo is available in a Github repository. Last time we saw how the inner product of two vectors gives rise to a decision rule: if $ w$ is the normal to a line (or hyperplane) $ L$, the sign of the inner product $ \langle x, w \rangle$ tells you whether $ x$ is on the same side of $ L$ as $ w$. ## The Inner Product as a Decision Rule DevFeed: [The Inner Product as a Decision Rule]() Original publisher: [Read original article]() Published: 2017-05-22T08:00:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [math](), [Code]() Tags: [arithmetic](), [inner-product](), [javascript](), [linear-algebra](), [machine-learning](), [math](), [mathematics](), [programming](), [projection](), [support-vector-machines](), [vectors]() ### AI overview This article explains how the standard inner product, or dot product, acts as a geometric decision rule. It shows how the sign of the product determines which side of a line through the origin a vector lies on, while zero indicates that the vector lies on the line. The explanation connects this behavior to vector projection. ### Source excerpt The standard inner product of two vectors has some nice geometric properties. Given two vectors $ x, y \in \mathbb{R}^n$, where by $ x_i$ I mean the $ i$-th coordinate of $ x$, the standard inner product (which I will interchangeably call the dot product) is defined by the formula $$\displaystyle \langle x, y \rangle = x_1 y_1 + \dots + x_n y_n$$ This formula, simple as it is, produces a lot of interesting geometry. ## Eigenfaces, for Facial Recognition DevFeed: [Eigenfaces, for Facial Recognition]() Original publisher: [Read original article]() Published: 2011-07-27T18:41:01Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Facial recognition](), [Algorithm]() Tags: [algorithm](), [covariance](), [eigenfaces](), [facial-recognition](), [images](), [inner-product](), [linear-algebra](), [mathematics](), [pixel](), [programming](), [vector]() ### AI overview This tutorial introduces eigenfaces for facial recognition. It outlines collecting sample face images, training a recognition algorithm, and classifying new images, then explains how grayscale images can be represented as matrices and vectors in a high-dimensional face space. ### Source excerpt This post assumes familiarity with the terminology and notation of linear algebra, particularly inner product spaces. Fortunately, we have both a beginner's primer on linear algebra and a follow-up primer on inner products. The Quest We are on a quest to write a program which recognizes images of faces. The general algorithm should be as follows. Get a bunch of sample images of people we want to recognize. Train our recognition algorithm on those samples. ## Inner Product Spaces--A Primer DevFeed: [Inner Product Spaces--A Primer]() Original publisher: [Read original article]() Published: 2011-07-25T00:29:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Computing]() Tags: [big-o-notation](), [eigenvectors](), [inner-product](), [mathematics](), [matrices](), [orthogonality](), [symmetry](), [vector-spaces](), [vectors]() ### AI overview This primer explains why vector spaces can be extended with a dot-product-like operation and defines inner products through conjugate symmetry, linearity, additivity, and positive definiteness. It introduces inner product spaces and discusses their connection to linear functionals. ### Source excerpt Vector spaces alone are not enough to do a lot of the interesting things we'd like them to do. Since a vector space is a generalization of Euclidean space, it is natural for us to investigate more specific types of vector spaces which are more akin to Euclidean space. In particular, we want to include the notion of a dot product. By admitting additional structure to a vector space, we may perform more computations, and hopefully get more interesting results.