# Questions tagged [inner-product]

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### Invariance signature in infinite dimension

Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that: $g_0$ is positive-definite $g_t$ is non-degenerate for ...
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### Concentration inequality for inner product of two Lipschitz functions

I was reading chapter 5 of the book HDP(Roman Vershynin). There I find theorem 5.1.4 extremely fascinating. I am curious to discover does this theorem hold to the inner product of two Lipschitz ...
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### Orthogonal functions on circle with constraints

I have a curious question I stumbled upon that may be interesting to some. Consider real-valued continuous functions on the circle $f_1(x),f_2(x),f_3(x)$ (so they are periodic in $x \mapsto x+2\pi$). ...
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### Largest number of two series vectors with negative dot product

Assume $\{x_{i}\}_{i=1}^{m}$, $\{w_{i}\}_{i=1}^{m}$ are two sets of vectors in $\mathbb{R}^{n}$. And we have that $x_{i}\cdot w_{j} < 0$ for $i \neq j$ and $x_{i}\cdot w_{i} > 0$ for all $i$. I ...
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### What is lost after RKHS embedding of the L1 space?

We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...
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### Inner products on super vector spaces

Let $V=V^0\oplus V^1$ be a super vector space (https://en.wikipedia.org/wiki/Super_vector_space) Is there a special definition of an inner product on $V$ other than just an inner product on the ...
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### Kernels with finite dimensional feature spaces

Suppose $x,y \in \mathbb{R}^n$ for some given fixed n. Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words,...
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### Matrix norm minimization and matrix inner product

One of the famous problem in SDP is the matrix norm minimization (see S. Boyd, Convex Optimization, p. 170). Consider: \label{eq:Lasse} \begin{aligned} &\min_{\mathbf{x}} & &...
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### A linear algebra question for semi-Euclidean norm

Let us consider Minkowski inner product on $\mathbb R^{1+n}$, defined by $$\langle v,w \rangle = -v_0w_0+\sum_{j=1}^n v_j w_j\quad \,\forall\, v,w \in \mathbb R^{1+n}.$$ We say that a vector $v$ is ...
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### Which inner products preserve positive correlation?

Suppose we have a symmetric PD or PSD matrix M which induces an inner product $\langle \cdot, \cdot \rangle_M$. If we have that $\langle x, y \rangle > 0$ for two unit vectors $x$, $y$, are there ...
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### Any inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative?

For some numerical analysis of a fluid, I am wondering if there is any inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In ...
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### On a special type of normed linear spaces

Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying $$\|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z}$$ is a group ...
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Denote $\mathcal I_m=\{-m,-m+1,\dots,0,\dots,m-1,m\}$ where $m\geq0$ is an integer. Pick a uniformly random vectors $$\hat a=(a_n,a_{n-1},\dots,a_1)\in(\mathcal I_{m_1}+\sqrt{-1}\cdot\mathcal I_{m_2}... • 13.1k 0 votes 0 answers 256 views ### Inner products on abelian groups and general modules Where can I find a discussion about inner products on something more general than vector spaces, and to which extent should one attempt such a generalization? My particular interest is in abelian ... 4 votes 0 answers 105 views ### Triple Petersson Inner Products With Theta Functions Our current work requires us to bound triple products of the form$$\langle \Im(*)^{\frac{3}{2}}|\theta^3|^2,\mu_j \rangle,$$where \langle \cdot, \cdot \rangle is the Petersson inner product, ... • 41 1 vote 0 answers 34 views ### Minimum element estimation of A*A^T [closed] I want to estimate the minimum element of A*A^T using only the information of A. For example, maximum element can be estimated using the norm values of all rows without calculating all elements of ... • 11 9 votes 1 answer 572 views ### Inner product over finite fields Let F be a finite field, For every c \in F, let X_1, X_2,..., X_9, Y_1,..., Y_9 be independent non-zero random variables over F. Denote X=(X_1,...,X_9), Y=(Y_1,...,Y_9), also let \... 2 votes 1 answer 220 views ### Invariant characterization of isometric embeddings \newcommand{\Cof}{\operatorname{Cof}} \newcommand{\Det}{\operatorname{Det}} \newcommand{\Lam}{\operatorname{\Lambda}} Motivation (and the "classic" case): I am trying to find a coordinate-free ... • 6,387 0 votes 1 answer 147 views ### Non-strict column diagonally dominant matrix inner product Let A \in \mathbb{R}^{n \times n} be a normalized non-strict column diagonally dominant matrix, that is:$$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$where 0 \le a_{j,j} \le 1 and -1 \le ... • 313 2 votes 0 answers 94 views ### How to calculate \langle v,w\rangle based only on \langle v,x_i\rangle and \langle w,x_i\rangle? [closed] Let's assume v,w, x_i \in R^n are unknown. Can one compute dot product \langle v,w\rangle if one has just the numbers: \langle v,x_i\rangle and \langle w,x_i\rangle for n random vectors ... • 255 1 vote 0 answers 464 views ### Complex conjugate and unitary complex conjugate Definition: Let V be complex finite dimensional inner product space Complex Conjugate: J is called complex conjugate on V iff (i) J is antilinear (ii) J^2 =I Definition: Anti-unitary Complex ... • 121 0 votes 2 answers 2k views ### Different inner products for vector spaces of random variables The inner product that appears in most books on probability is the covariance \langle X,Y \rangle = E[XY] (considering that X and Y are zero mean real random variables). Are there other inner ... 4 votes 1 answer 336 views ### Existence of SO(n)-isotropic inner products which are not O(n)-isotropic \newcommand{\al}{\alpha} Let M_n be the space of n \times n real matrices. Question: For which n, is there an inner product on M_n which satisfies:$$(*) \, \, \langle Q^TXQ,Q^TYQ \...
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Consider a set of vectors $M$ from an inner product space $V$. The ordered set of inner products of all pairs of elements in $M$ uniquely characterizes $M$ up to isomorphism. Suppose now that $V$ is ...