{"author":{"avatar":"https://nguyencungthanh.github.io/animation_avatar%282%29.PNG","name":"Nguyen Cung Thanh","url":"https://nguyencungthanh.github.io/"},"feed_url":"https://nguyencungthanh.github.io/tags/feature-map/feed.json","home_page_url":"https://nguyencungthanh.github.io/tags/feature-map/","items":[{"content_html":"\n  \u003cdiv class='e-content'\u003e\n    \u003ccenter\u003e\u003cb\u003eAbstract\u003c/b\u003e\u003c/center\u003e\n\u003cp\u003e$\\quad$ We study representations of functions of inner products via embeddings into Hilbert spaces. While polynomials with nonnegative coefficients admit a single feature map realization, this approach fails when negative coefficients are present. To address this limitation, we introduce a two-map framework. Specifically, for any analytic function $f(x) = \\sum_{k=0}^{\\infty} a_k x^k$ with global convergence, we construct a Hilbert space $\\mathcal{H}$ and transformations $\\Phi, \\Psi: \\mathbb{R}^n \\to \\mathcal{H}$ such that\n$$\\langle \\Phi(u), \\Psi(v) \\rangle = f(\\langle u, v \\rangle)\n\\quad \\text{for all } u, v \\in \\mathbb{R}^n.$$\u003c/p\u003e\n\u003ch2 id=\"1-real-analytic-function\"\u003e1. Real analytic function\u003c/h2\u003e\n\u003cp\u003eWe start with the following problem:\u003c/p\u003e\n\u003cp\u003eLet $P(x)$ be a polynomial with nonnegative coefficients, then there exists a Hilbert space $\\mathcal{H}$ and a transformation $\\Phi : \\mathbb{R}^n \\to \\mathcal{H}$ such that:\n$$\\langle \\Phi(u), \\Phi(v) \\rangle = P(\\langle u, v \\rangle) \\quad \\text{for all } u, v \\in \\mathbb{R^n}.$$\nBut our task can be impossible if $P$ has a negative coefficient since $\\langle \\Phi(u), \\Phi(v) \\rangle$ is always nonnegative. Therefore, a natural question is what if we consider two transformations $\\Phi, \\Psi: \\mathbb{R}^n \\to \\mathcal{H}?$ This leads us to a more general problem about analytic function as follow.\u003c/p\u003e\n\u003cp\u003eConsider a function\n$$\nf(x) = \\sum_{k=0}^{\\infty} a_k x^k\n$$\nwhere the series absolutely converges for all $x \\in \\mathbb{R}$. There exist a Hilbert space $\\mathcal{H}$ and transformations $\\Phi, \\Psi : \\mathbb{R}^n \\to \\mathcal{H}$ such that\n$$\n\\langle \\Phi(u), \\Psi(v) \\rangle = f(\\langle u, v \\rangle)\n\\quad \\text{for all } u, v \\in \\mathbb{R}^n.\n$$\nAlso, for any unit vector $u$, we have\n$$ \\|\\Phi(u)\\|^2_\\mathcal{H} = \\|\\Psi(u)\\|^2_\\mathcal{H} =\\sum _{k=0}^{\\infty} |a_k| .$$\n\u003cem\u003eProof.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eFor each $k \\ge 0$, let $V_k = \\mathbb{R}^{n^k}$ and define\n$$\nH = \\bigoplus_{k=0}^{\\infty} V_k =\n\\left\\{ (w_0, w_1, \\dots) : w_k \\in V_k,\\ \\sum_{k=0}^{\\infty} \\|w_k\\|^2 \u0026lt; \\infty \\right\\}\n$$\u003c/p\u003e\n\u003cp\u003ewith inner product\n$$\n\\langle w, z \\rangle_\\mathcal{H} = \\sum_{k=0}^{\\infty} \\langle w_k, z_k \\rangle_{V_k}.\n$$\u003c/p\u003e\n\u003cp\u003eConsider $s_k = \\operatorname{sign}(a_k)$ and write $a_k = s_k |a_k|$. For $u \\in \\mathbb{R}^n$, define\n$$\n\\Phi(u) = (\\sqrt{|a_0|} u^{\\otimes 0}, \\sqrt{|a_1|} u^{\\otimes 1}, \\dots),\n$$\u003c/p\u003e\n\u003cp\u003e$$\n\\Psi(v) = (s_0 \\sqrt{|a_0|} v^{\\otimes 0}, s_1 \\sqrt{|a_1|} v^{\\otimes 1}, \\dots).\n$$\u003c/p\u003e\n\u003cp\u003eThen we prove that $\\Phi(u), \\Psi(v) \\in \\mathcal{H}$. Indeed, we have:\n$$\n\\left\\|\\sqrt{|a_k|} u^{\\otimes k}\\right\\|^2 = |a_k| \\left\\|u^{\\otimes k}\\right\\|^2 = |a_k| \\|u\\|^{2k}.\n$$\u003c/p\u003e\n\u003cp\u003eThus,\n$$ \\sum_{k=0}^{\\infty}\\left\\|\\sqrt{|a_k|} u^{\\otimes k}\\right\\|^2 = \\sum_{k=0}^{\\infty} |a_k| \\|u\\|^{2k}\n$$\nconverges, so $\\Phi(u) \\in \\mathcal{H}$. Similarly, $\\Psi(v) \\in \\mathcal{H}$.\u003c/p\u003e\n\u003cp\u003eOn the other hand,\n$$\n\\begin{aligned}\n\\langle \\Phi(u), \\Psi(v) \\rangle_\\mathcal{H} \u0026amp;=\n\\sum_{k=0}^{\\infty} \\left\\langle \\sqrt{|a_k|} u^{\\otimes k}, s_k \\sqrt{|a_k|} v^{\\otimes k}\n\\right\\rangle_{V_k} \\\\\n\u0026amp;= \\sum_{k=0}^{\\infty} s_k |a_k| \\langle u^{\\otimes k}, v^{\\otimes k} \\rangle\\\\\n\u0026amp;= \\sum_{k=0}^{\\infty} a_k \\langle u, v \\rangle^k \\\\\n\u0026amp;= f(\\langle u, v \\rangle).\n\\end{aligned}\n$$\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eNote.\u003c/strong\u003e Norm when $u$ is a unit vector: If $\\|u\\| = 1$, then\n$$\n\\|u^{\\otimes k}\\|^2 = \\|u\\|^{2k} = 1.\n$$\u003c/p\u003e\n\u003cp\u003eHence,\n$$\\|\\Phi(u)\\|_\\mathcal{H}^2 = \\sum _{k=0}^{\\infty} |a_k| \\| u^{\\otimes k} \\|^2 = \\sum _{k=0}^{\\infty} |a_k|.$$\u003c/p\u003e\n\u003ch3 id=\"example\"\u003eExample\u003c/h3\u003e\n\u003cp\u003e\u003cstrong\u003eSine.\u003c/strong\u003e  Let $c \u0026gt; 0$. Then function $ f(x) = \\sin(cx) $ is real analytic\n$$\n\\sin(cx) = cx - \\frac{(cx)^3}{3!} + \\frac{(cx)^5}{5!} - \\ldots\n$$\u003c/p\u003e\n\u003cp\u003eThus, there exists a Hilbert space $\\mathcal{H}$ and transformation $\\Phi, \\Psi: \\mathbb{R}^n \\to \\mathcal{H}$ such that\n$$\n\\langle \\Phi(u), \\Psi(v) \\rangle = \\sin(c \\langle u, v \\rangle)\\quad \\text{for all } u, v \\in \\mathbb{R}^n.\n$$\u003c/p\u003e\n\u003cp\u003eAlso, $\\Phi$ and $\\Psi$ map unit vectors to unit vectors if\n$$\n1 = c + \\frac{c^3}{3!} + \\frac{c^5}{5!} + \\ldots = \\frac{e^c - e^{-c}}{2}.\n$$\u003c/p\u003e\n\u003cp\u003eHence,\n$$\nc = \\ln(1 + \\sqrt{2}).\n$$\u003c/p\u003e\n\u003ch2 id=\"2-kernels-and-feature-maps\"\u003e2. Kernels and feature maps\u003c/h2\u003e\n\u003cp\u003eGiven a function of two variables $ K : \\mathcal{X} \\times \\mathcal{X} \\to \\mathbb{R},$ on some set sets $\\mathcal{X}$, when can we find a Hilbert space $\\mathcal{H}$ and a transformation $\\Phi : \\mathcal{X} \\to \\mathcal{H}$ so that\u003c/p\u003e\n\u003cspan class=\"sidenote\" id=\"note1\"\u003e\n    1. The kernel trick, which expresses a kernel $K(u, v)$ as an inner product, is widely used in machine learning because it lets us handle non-linear models (determined by $K$) with techniques designed for linear models.\n\u003c/span\u003e\n\u003cp\u003e$$\n\\langle \\Phi(u), \\Phi(v) \\rangle = K(u,v)^{\\textcolor{gray} 1}  \\quad \\text{for all } u, v \\in \\mathcal{X}?\n$$\u003c/p\u003e\n\u003cp\u003eThe answer is given by Mercer theorem\u003csup\u003e2\u003c/sup\u003e\n\u003cspan class=\"sidenote\" id=\"note2\"\u003e\n2. \u003ca href=\"https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/11/Mercer-original-compressed_compressed.pdf\"\u003e Functions of positive and negative type and their connection with the theory of integral equations\u003c/a\u003e (Mercer, 1909).\n\u003c/span\u003e and, more precisely, Moore-Aronszajn theorem:\u003c/p\u003e\n\u003cp\u003e$-$ The necessary and sufficient condition: $K$ is a \u003cem\u003epositive semidefinite kernel\u003c/em\u003e, i.e., for any $u_1, \\dots, u_N \\in \\mathcal{X}$, the matrix\n$ (K(u_i, u_j))_{i,j=1}^N $ is symmetric and positive semidefinite.\u003c/p\u003e\n\u003cp\u003e$-$ Transformation $\\Phi$ is called a \u003cem\u003efeature map\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003e$-$ The Hilbert space $\\mathcal{H}$ is called a \u003cem\u003ereproducing kernel Hilbert space\u003c/em\u003e\u003csup\u003e3\u003c/sup\u003e (RKHS).\n\u003cspan class=\"sidenote\" id=\"note3\"\u003e\n3. \u003ca href=\"https://arxiv.org/abs/2106.08443\"\u003e Reproducing Kernel Hilbert Space, Mercer\u0026rsquo;s Theorem, Eigenfunctions, Nyström Method, and Use of Kernels in Machine Learning: Tutorial and Survey\u003c/a\u003e (Ghojogh et al, 2021).\n\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003e$-$ Popular positive semidefinite kernels in machine learning include the Gaussian and polynomial kernels given by:\n$$\nK(u,v) = \\exp \\left( -\\frac{\\|u - v\\|^2}{2\\sigma^2} \\right), \\quad K(u,v) = (\\langle u, v \\rangle + r)^k, \\quad u, v \\in \\mathbb{R}^n.\n$$\nwhere $\\sigma \u0026gt; 0, r \u0026gt; 0,$ and $k \\in \\mathbb{N}$ are parameters.\u003c/p\u003e\n\u003ch2 id=\"references\"\u003eReferences\u003c/h2\u003e\n\u003cp\u003e[1] Vershynin, R. (2026). \u003cem\u003eHigh-dimensional probability: An introduction with applications in data science\u003c/em\u003e (2nd ed.). Cambridge University Press.\u003c/p\u003e\n  \u003c/div\u003e\n\u003cp\u003e\n  \u003ca href=\"mailto:nguyencungthanh11@gmail.com?subject=Re: Real%20Analytic%20Function%20in%20Hilbert%20space\" rel='noopener noreferrer'\u003e💬 Reply via e-mail\u003c/a\u003e\n\u003c/p\u003e\n","date_modified":"2026-04-29T00:00:00+00:00","date_published":"2026-04-29T00:00:00+00:00","id":"https://nguyencungthanh.github.io/2026/04/29/real-analytic-function-in-hilbert-space/","summary":"Abstract $\\quad$ We study representations of functions of inner products via embeddings into Hilbert spaces. While polynomials with nonnegative coefficients admit a single feature map realization, this approach fails when negative coefficients are present. To address this limitation, we introduce a two-map framework. Specifically, for any analytic function $f(x) = \\sum_{k=0}^{\\infty} a_k x^k$ with global convergence, we construct a Hilbert space $\\mathcal{H}$ and transformations $\\Phi, \\Psi: \\mathbb{R}^n \\to \\mathcal{H}$ such that $$\\langle \\Phi(u), \\Psi(v) \\rangle = f(\\langle u, v \\rangle) \\quad \\text{for all } u, v \\in \\mathbb{R}^n.$$\n","tags":["Functional Analysis","Tensor","Feature Map"],"title":"Real Analytic Function in Hilbert space","url":"https://nguyencungthanh.github.io/2026/04/29/real-analytic-function-in-hilbert-space/"}],"title":"Tag: Feature Map - Nguyen Cung Thanh","version":"https://jsonfeed.org/version/1"}