Gaussian Hilbert Space Overview

• Gaussian Hilbert space is a mathematical framework defined by a Hilbert space equipped with Gaussian measures for analyzing infinite-dimensional random phenomena.
• It rigorously characterizes covariance operators, geometric metrics, and regularized divergences, facilitating advances in stochastic processes and quantum state analysis.
• Its practical applications span quantum theory, kernel methods in machine learning, and statistical inference, enabling deeper insights into complex probabilistic systems.

A Gaussian Hilbert space is a complex mathematical structure at the intersection of probability theory, functional analysis, and quantum theory, formalized as a Hilbert space equipped with Gaussian probabilistic or operator-theoretic features. This concept encompasses several fundamental aspects: the structure of infinite-dimensional Gaussian measures, Gaussian states in quantum mechanics, covariance operators and their associated geometry, and the analytical apparatus required for stochastic processes and quantum fields.

1. Structure of Gaussian Measures in Hilbert Spaces

A Gaussian measure on a separable Hilbert space $H$ is a probability measure under which every continuous linear functional has a (real) Gaussian distribution. Concretely, a Borel measure $\mu$ on $H$ is Gaussian if for every $f \in H^*$ (with the Riesz representation identifying $H^* \simeq H$), the map $x \mapsto \langle x, f \rangle_H$ has a normal distribution under $\mu$.

A Gaussian measure $\mu = N(m, C)$ on $H$ is determined by:

• Mean $m \in H$.
• Covariance operator $C \in L^+(H)$ (self-adjoint, positive, trace class), such that

$$\int_H \langle x - m, h_1 \rangle_H \langle x - m, h_2 \rangle_H \, d\mu(x) = \langle C h_1, h_2 \rangle_H, \quad \forall h_1, h_2 \in H.$$

In infinite dimensions, such measures are never equivalent to Lebesgue measure (which does not exist), and many properties—such as densities and the notion of singularity—differ fundamentally from the finite-dimensional case.

A pivotal result is the Feldman-Hajek theorem: two Gaussian measures $N(m_1, C_1)$, $N(m_2, C_2)$ on $H$ are equivalent if and only if $m_2 - m_1 \in \operatorname{Im}(C_1^{1/2})$ and the covariance operators are equivalent in a sense involving Hilbert-Schmidt perturbations (see (2506.10494); also (1904.05352)).

2. Covariance Operators and the Geometry of Gaussian States

The covariance operator plays a central role in both the measure-theoretic and operator-theoretic structure.

• For a centered Gaussian measure $N(0, C)$ on $H$ (with $C$ strictly positive and trace class), all moments are determined by $C$.
• The geometry of the space of covariance operators is rich and can be analyzed through the lens of Riemannian geometry.

Fisher-Rao Geometry

The space of equivalent centered Gaussian measures on $H$ forms a Hilbert (infinite-dimensional) manifold, with the Fisher-Rao metric generalizing its finite-dimensional form (2310.10182): $$g_\Sigma(A, B) = \frac{1}{2} \operatorname{tr}( \Sigma^{-1} A \Sigma^{-1} B )$$ where $\Sigma$ is a (positive definite, trace class) covariance operator, and $A, B$ are tangents (symmetric, Hilbert-Schmidt operators).

• Geodesics: The unique geodesic connecting $\Sigma_0$ and $\Sigma_1$ is:

$$\gamma(t) = \Sigma_0^{1/2} \exp( t \log( \Sigma_0^{-1/2} \Sigma_1 \Sigma_0^{-1/2} ) ) \Sigma_0^{1/2}, \quad t \in [0,1].$$

• Riemannian distance:

$$d(\Sigma_0, \Sigma_1) = \frac{1}{\sqrt{2}} \| \log( \Sigma_0^{-1/2} \Sigma_1 \Sigma_0^{-1/2} ) \|_{HS}$$

where $\| \cdot \|_{HS}$ denotes the Hilbert-Schmidt norm.

3. Gaussian States and Operator Algebras

In quantum theory, Gaussian states arise as elements in the Hilbert space $L^2(\mathbb{R}^n)$, or more generally, Fock spaces, associated to canonical position and momentum observables (1101.5041). A Gaussian state is a quantum state such that all real linear combinations of the canonical observables $q_j, p_j$ (for $1 \leq j \leq n$) have a joint normal distribution.

The set $S_n$ of all Gaussian states is invariant under the action of:

• Symplectic group $Sp(2n, \mathbb{R})$: representing linear canonical (phase-space) transformations preserving the symplectic form.
• Weyl operators: implementing translations (displacements) in phase space.

A key result is that all unitary symmetries preserving $S_n$ are of the form

$U = \lambda W(\bm{\alpha}) \Gamma(L)$

where $W(\bm{\alpha})$ encodes displacements, $\Gamma(L)$ implements the Bogolioubov (symplectic) automorphism, and $\lambda$ is a phase (1101.5041).

4. Divergences and Information Geometry in Gaussian Hilbert Spaces

Divergences between Gaussian measures/general quantum states in Hilbert space are central in statistics, information geometry, and quantum information theory. In infinite dimensions, these divergences require regularization and operator theory for rigorous definition.

Kullback-Leibler, Rényi, and Log-Determinant Divergences

• The Kullback-Leibler (KL) divergence between two Gaussians $N(m_1, C_1)$, $N(m_2, C_2)$ is (regularized):

$$D^{\gamma}_{\rm KL}(N(m_1, C_1) \| N(m_2, C_2)) = \frac{1}{2} \langle m_1 - m_2, (C_2 + \gamma I)^{-1}(m_1 - m_2) \rangle + \frac{1}{2} d^1[C_1+\gamma I, C_2+\gamma I]$$

where $d^1$ denotes a regularized log-determinant divergence (1904.05352).

• Rényi divergences and Alpha Log-Determinant divergences extend these concepts using regularization and trace/Hilbert-Schmidt class operators (2207.08406).
• As $\gamma \to 0$, these regularized divergences converge to their "true" values when the measures are equivalent.

Geometric Jensen-Shannon Divergence

The Geometric Jensen-Shannon divergence (GJS) is defined for equivalent Gaussian measures $N(m_0, C_0), N(m_1, C_1)$ in an infinite-dimensional Hilbert space as (2506.10494): $$\mathrm{JS}_{G_\alpha}(N(m_0, C_0) \| N(m_1, C_1)) = (1-\alpha) \mathrm{KL}(N(m_0, C_0) \| N(m_\alpha, C_\alpha)) + \alpha \mathrm{KL}(N(m_1, C_1) \| N(m_\alpha, C_\alpha)),$$ where $N(m_\alpha, C_\alpha)$ is the geometric mean (interpolation) of the two measures, with covariance and mean

$$C_\alpha = [(1-\alpha) C_0^{-1} + \alpha C_1^{-1} ]^{-1}, \qquad m_\alpha = C_\alpha [ (1-\alpha) C_0^{-1} m_0 + \alpha C_1^{-1} m_1 ].$$

In general, for arbitrary (not necessarily equivalent) measures, a regularized GJS is defined via addition of $\gamma I$ to the covariance operators, ensuring the expressions are trace class and determinants are well-defined (2506.10494).

5. Conditioning and Marginals: The Shorted Operator

For a Gaussian measure on a Hilbert space $H$ with covariance $C$ and subspace $S$, conditioning on the complement $S^\perp$ yields a Gaussian measure on $S$ with covariance equal to the shorted operator $\mathcal{S}(C)$ (1506.04208). This generalizes the notion of the Schur complement to infinite dimensions and is vital for infinite-dimensional Bayesian inference.

Given block decomposition $H = S \oplus S^\perp$, if $C$ has blocks

$$C = \begin{pmatrix} C_{SS} & C_{S S^\perp} \ C_{S^\perp S} & C_{S^\perp S^\perp} \end{pmatrix},$$

then, when invertible,

$$\mathcal{S}(C) = C_{SS} - C_{S S^\perp} C_{S^\perp S^\perp}^{-1} C_{S^\perp S},$$

with generalizations via variational characterization and approximation sequences for non-invertible or infinite-dimensional cases.

6. Gaussian Hilbert Spaces and Functional Analysis

In functional analysis, a Gaussian Hilbert space may refer to a Hilbert space of square-integrable random variables (with respect to a Gaussian measure), or the RKHS associated to a Gaussian process or kernel (e.g., the Cameron-Martin space of a Wiener process).

The structure of function spaces with Gaussian kernels—e.g., in machine learning (reduced-rank GP regression, kernel quadrature)—relies crucially on spectral decompositions and Hilbert space geometry (1401.5508, 2004.11408). These methods connect the sampling, approximation, and integration properties of Gaussian fields to their spectral and kernel-theoretic properties.

7. Universality, Positive Definiteness, and Infinite Dimensionality

The Gaussian kernel $k(x, y) = \exp(-\sigma\|x-y\|^2)$ is universal, strictly positive definite, and integrally strictly positive definite on any real Hilbert space, including infinite dimensions (2007.14697). Universality here means the RKHS is dense in the space of continuous or vanishing-at-infinity functions, which underpins its broad applicability in approximation, learning, and statistics.

Summary Table: Key Objects and Formulas

Aspect	Key Formula / Property	Source
Covariance of Gaussian on $H$	$Cx = \int_H (y - m, x)(y - m)\, d\mu(y)$	(2506.10494)
Fisher-Rao metric	$$g_\Sigma(A, B) = \frac{1}{2} \operatorname{tr}(\Sigma^{-1} A \Sigma^{-1} B)$$	(2310.10182)
GJS divergence (Hilbert space)	See formula in Section 2.3 above	(2506.10494)
Shorted operator	$$\mathcal{S}(C) = C_{SS} - C_{S S^\perp} C_{S^\perp S^\perp}^{-1} C_{S^\perp S}$$	(1506.04208)
RKHS universality of Gaussian kernel	$G_\sigma$ is universal and ISPD on any real Hilbert space	(2007.14697)
Hilbert space reduced-rank GP expansion	$$k(x,x') \approx \sum_{j=1}^m S(\sqrt{\lambda_j})\phi_j(x)\phi_j(x')$$	(1401.5508, 2004.11408)
Condition for paths in RKHS	$\sum_{i} \sqrt{\mu_i(T_{X,\nu})}<\infty$	(2407.11898)
Regularized KL divergence	See formula with $D^{\gamma}_{\mathrm{KL}}$ in Section 4	(1904.05352)

Conclusion

The theory of Gaussian Hilbert space unifies deep operator-theoretic, probabilistic, geometric, and computational aspects, underpinning much of modern infinite-dimensional probability, stochastic analysis, kernel methods, and quantum theory. The central objects—Gaussian measures, covariance operators, regularized divergences, geometric structures, and spectral decompositions—are linked by analytically tractable formulas and geometric principles, supporting broad developments in both pure mathematics and applications in statistics, machine learning, and quantum information.