Projected Isotropic Normal Distribution

• The projected isotropic normal distribution is defined by projecting an isotropic Gaussian vector onto subspaces or manifolds (e.g., spheres), preserving rotational invariance.
• It underpins key applications in directional statistics, multivariate visualization, and manifold inference by providing exact moment formulas and identifiable parameter structures.
• Recent studies extend the model to high-dimensional settings and random modulation, revealing convergence to normal variance-mixtures and enabling robust statistical inference.

Searching arXiv for papers on the projected isotropic normal distribution and closely related projected normal models. The projected isotropic normal distribution is the distribution obtained by projecting an isotropic Gaussian random vector onto a lower-dimensional linear subspace or onto a unit sphere, circle, or hypertorus, depending on the projection map. In the Euclidean linear-projection setting, an isotropic Gaussian $X \sim \mathcal{N}(0,\sigma^2 I_d)$ remains Gaussian under every orthogonal projection, so any fixed one-dimensional projection is exactly $\mathcal{N}(0,\sigma^2)$ and any $k$-dimensional orthogonal projection is $\mathcal{N}(0,\sigma^2 I_k)$ (Chaudhury, 2011, Calvi et al., 4 Feb 2025). In the directional setting, the normalized variable $Y=X/\|X\|$ lies on the unit sphere and follows a projected normal, also called an angular Gaussian; under isotropy and zero mean this reduces to the uniform law on the sphere, whereas nonzero mean induces a symmetric, unimodal directional law determined by the ratio of mean magnitude to isotropic noise scale (Herrera-Esposito et al., 20 Jun 2025, Figueras et al., 2024, Mardia et al., 4 Mar 2026). Recent work also studies random linear modulation $Y_n=\Xi_n'X_n$, where isotropic normal inputs and spherically symmetric modulators generate exact or asymptotic normal variance-mixtures, together with quantitative convergence, matrix-normal limits, and characterizations of when the modulator itself must be Gaussian (Bagyan et al., 14 Oct 2025).

1. Euclidean definition and isotropic projection principle

In its most basic form, the projected isotropic normal begins with an isotropic Gaussian random vector

$X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$

The isotropy condition means that the covariance is $\sigma^2 I_d$, so the law is rotationally invariant around its mean when $\mu=0$ (Chaudhury, 2011, Calvi et al., 4 Feb 2025). If $P\in\mathbb{R}^{k\times d}$ has orthonormal rows, $\mathcal{N}(0,\sigma^2)$0, and one defines the linear projection

$\mathcal{N}(0,\sigma^2)$1

then

$\mathcal{N}(0,\sigma^2)$2

In the isotropic case $\mathcal{N}(0,\sigma^2)$3, this simplifies to

$\mathcal{N}(0,\sigma^2)$4

hence every linear projection is again isotropic normal in the projected subspace (Calvi et al., 4 Feb 2025).

Several immediate consequences are standard. For any unit vector $\mathcal{N}(0,\sigma^2)$5,

$\mathcal{N}(0,\sigma^2)$6

and in the zero-mean isotropic case,

$\mathcal{N}(0,\sigma^2)$7

identical for every unit-length direction by isotropy (Bagyan et al., 14 Oct 2025, Chaudhury, 2011). Likewise, an orthogonal projection onto any $\mathcal{N}(0,\sigma^2)$8-dimensional subspace yields $\mathcal{N}(0,\sigma^2)$9 on that image subspace (Chaudhury, 2011).

This Euclidean notion should be distinguished from directional projection. In linear projection, the image remains a Gaussian distribution in $k$0; in spherical or circular projection, the vector is normalized by its norm and the image lies on a manifold such as $k$1 or $k$2 (Herrera-Esposito et al., 20 Jun 2025, Figueras et al., 2024). The shared terminology reflects the common geometric origin in isotropic Gaussian structure.

2. Radial–angular structure and projection to spheres

For $k$3, the radial–angular decomposition is especially simple. Writing $k$4 and $k$5 with $k$6, one has that $k$7 and $k$8 are independent, and $k$9 is uniform on $\mathcal{N}(0,\sigma^2 I_k)$0 (Chaudhury, 2011). The radius satisfies

$\mathcal{N}(0,\sigma^2 I_k)$1

and its density is the chi density

$\mathcal{N}(0,\sigma^2 I_k)$2

(Chaudhury, 2011).

From this viewpoint, isotropy means that the entire directional distribution is encoded by the radial law. When the mean vanishes, projecting onto the unit sphere produces no directional preference: the projected normal is exactly uniform (Chaudhury, 2011, Herrera-Esposito et al., 20 Jun 2025). When the mean is nonzero, the projected distribution becomes concentrated around the direction of the mean, but the isotropic covariance still enforces axisymmetry around that direction (Figueras et al., 2024, Mardia et al., 4 Mar 2026).

A constructive 2D limit mechanism is given by the radial central limit theorem. Let $\mathcal{N}(0,\sigma^2 I_k)$3 be supported on a line through the origin in the plane, with zero mean and unit variance, and define

$\mathcal{N}(0,\sigma^2 I_k)$4

with equally spaced angles $\mathcal{N}(0,\sigma^2 I_k)$5. Then $\mathcal{N}(0,\sigma^2 I_k)$6 converges in distribution to the standard normal distribution on the plane (Chaudhury, 2011). This yields a probabilistic interpretation of isotropic Gaussian structure from rotated one-dimensional components, and it clarifies why angular homogenization plus averaging produces a radial Gaussian limit.

In high dimensions, the isotropic Gaussian radius exhibits thin-shell concentration. Since $\mathcal{N}(0,\sigma^2 I_k)$7 with mean $\mathcal{N}(0,\sigma^2 I_k)$8 and variance $\mathcal{N}(0,\sigma^2 I_k)$9, the radius concentrates around $Y=X/\|X\|$0, with $Y=X/\|X\|$1 fluctuations rather than $Y=X/\|X\|$2 fluctuations (Chaudhury, 2011). This thin-shell behavior is directly connected to the high-dimensional projection limits studied for random modulation and conditional normality (Bagyan et al., 14 Oct 2025).

3. Directional projected isotropic normal on the circle and sphere

On the circle, the projected isotropic normal arises from

$Y=X/\|X\|$3

with phase angle

$Y=X/\|X\|$4

Writing $Y=X/\|X\|$5, the distribution depends on the mean direction $Y=X/\|X\|$6 and the scalar concentration parameter

$Y=X/\|X\|$7

equivalently $Y=X/\|X\|$8 (Mardia et al., 4 Mar 2026). Its exact density is

$Y=X/\|X\|$9

for $Y_n=\Xi_n'X_n$0, where $Y_n=\Xi_n'X_n$1 and $Y_n=\Xi_n'X_n$2 are the standard normal pdf and cdf (Mardia et al., 4 Mar 2026). Under isotropy, the density depends only on $Y_n=\Xi_n'X_n$3, is unimodal with mode at $Y_n=\Xi_n'X_n$4, and is symmetric:

$Y_n=\Xi_n'X_n$5

(Mardia et al., 4 Mar 2026).

An equivalent isotropic-circle parametrization uses $Y_n=\Xi_n'X_n$6 and $Y_n=\Xi_n'X_n$7, giving

$Y_n=\Xi_n'X_n$8

(Mastrantonio, 2017). The two formulations are consistent through $Y_n=\Xi_n'X_n$9.

On $X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$0, if $X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$1 and

$X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$2

then $X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$3 has a projected normal distribution on the sphere (Figueras et al., 2024). Its density with respect to surface area measure can be written as

$X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$4

and, in closed form,

$X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$5

with

$X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$6

(Figueras et al., 2024). As $X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$7, this tends to the uniform density $X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$8 on $X \sim \mathcal{N}_d(\mu,\sigma^2 I_d).$9 (Figueras et al., 2024).

A more general sphere-valued projected normal is defined by

$\sigma^2 I_d$0

with density on $\sigma^2 I_d$1 expressed through

$\sigma^2 I_d$2

and a recursively defined function $\sigma^2 I_d$3 (Herrera-Esposito et al., 20 Jun 2025). In the isotropic specialization $\sigma^2 I_d$4, these simplify to $\sigma^2 I_d$5, $\sigma^2 I_d$6, and $\sigma^2 I_d$7, and when $\sigma^2 I_d$8 the law becomes uniform on $\sigma^2 I_d$9 (Herrera-Esposito et al., 20 Jun 2025).

A common misconception is to identify the projected isotropic normal with the von Mises or von Mises–Fisher family. The cited works explicitly distinguish them: the projected isotropic normal arises from Euclidean Gaussian normalization, while the von Mises and von Mises–Fisher families are exponential-family models on compact manifolds (Figueras et al., 2024, Kato et al., 22 Aug 2025, Mardia et al., 4 Mar 2026). Approximation links exist, but the models are not the same.

4. Exact moments, intrinsic statistics, and identifiability

For the sphere-valued isotropic projected normal $\mu=0$0 with $\mu=0$1 in $\mu=0$2, exact isotropic moment formulas are available. The mean has the form

$\mu=0$3

where

$\mu=0$4

(Herrera-Esposito et al., 20 Jun 2025). The second moment is

$\mu=0$5

and the covariance can be written as

$\mu=0$6

with coefficients $\mu=0$7 and $\mu=0$8 given explicitly in terms of confluent hypergeometric functions (Herrera-Esposito et al., 20 Jun 2025). When $\mu=0$9, these formulas reduce to $P\in\mathbb{R}^{k\times d}$0 and $P\in\mathbb{R}^{k\times d}$1, the uniform case (Herrera-Esposito et al., 20 Jun 2025).

On $P\in\mathbb{R}^{k\times d}$2, intrinsic statistics provide a geometric formulation. If $P\in\mathbb{R}^{k\times d}$3 with $P\in\mathbb{R}^{k\times d}$4, the intrinsic Fréchet mean is

$P\in\mathbb{R}^{k\times d}$5

and under isotropy the expectation commutes with projection:

$P\in\mathbb{R}^{k\times d}$6

provided $P\in\mathbb{R}^{k\times d}$7 (Figueras et al., 2024). If $P\in\mathbb{R}^{k\times d}$8, the distribution is uniform on $P\in\mathbb{R}^{k\times d}$9 and the intrinsic mean is not unique (Figueras et al., 2024).

The intrinsic covariance on the tangent plane $\mathcal{N}(0,\sigma^2)$00 is isotropic:

$\mathcal{N}(0,\sigma^2)$01

where $\mathcal{N}(0,\sigma^2)$02 is a continuous, strictly increasing bijection from $\mathcal{N}(0,\sigma^2)$03 onto $\mathcal{N}(0,\sigma^2)$04 (Figueras et al., 2024). This yields a one-to-one correspondence between the scale-free Euclidean parameter $\mathcal{N}(0,\sigma^2)$05 and the intrinsic covariance of the projected spherical distribution (Figueras et al., 2024).

Identifiability is a central issue. Phase-only or direction-only data do not identify $\mathcal{N}(0,\sigma^2)$06 and $\mathcal{N}(0,\sigma^2)$07 separately; only their ratio is identifiable. On the circle, the identifiable parameters are $\mathcal{N}(0,\sigma^2)$08 and $\mathcal{N}(0,\sigma^2)$09 (Mardia et al., 4 Mar 2026). On $\mathcal{N}(0,\sigma^2)$10, the estimable quantities from projected data alone are the direction $\mathcal{N}(0,\sigma^2)$11 and the scale-free variance $\mathcal{N}(0,\sigma^2)$12 (Figueras et al., 2024). More generally, the projected normal inherits a scale non-identifiability: scaling a Gaussian pair $\mathcal{N}(0,\sigma^2)$13 by a positive constant leaves the angle unchanged (Mastrantonio, 2017). Bayesian treatments therefore impose identification constraints such as fixing one variance per circular pair and post-processing posterior samples accordingly (Mastrantonio, 2017).

For the circular projected isotropic normal, recent work derives exact trigonometric moments. If $\mathcal{N}(0,\sigma^2)$14, then

$\mathcal{N}(0,\sigma^2)$15

and, for integer $\mathcal{N}(0,\sigma^2)$16,

$\mathcal{N}(0,\sigma^2)$17

(Mardia et al., 4 Mar 2026). In particular,

$\mathcal{N}(0,\sigma^2)$18

and

$\mathcal{N}(0,\sigma^2)$19

(Mardia et al., 4 Mar 2026). These closed forms make the mean resultant and its square analytically accessible.

5. Random projections, variance-mixtures, and high-dimensional modulation

A distinct but closely related regime considers scalar projections formed by random modulation:

$\mathcal{N}(0,\sigma^2)$20

where $\mathcal{N}(0,\sigma^2)$21 are independent (Bagyan et al., 14 Oct 2025). When

$\mathcal{N}(0,\sigma^2)$22

the conditional law is exact:

$\mathcal{N}(0,\sigma^2)$23

In the zero-mean isotropic case,

$\mathcal{N}(0,\sigma^2)$24

(Bagyan et al., 14 Oct 2025).

Thus the unconditional law is a normal variance-mixture with mixing variable

$\mathcal{N}(0,\sigma^2)$25

Its density and characteristic function are

$\mathcal{N}(0,\sigma^2)$26

$\mathcal{N}(0,\sigma^2)$27

(Bagyan et al., 14 Oct 2025). This statement is exact for isotropic Gaussian $\mathcal{N}(0,\sigma^2)$28 and independent spherically symmetric $\mathcal{N}(0,\sigma^2)$29.

If $\mathcal{N}(0,\sigma^2)$30 is itself Gaussian, $\mathcal{N}(0,\sigma^2)$31, and $\mathcal{N}(0,\sigma^2)$32 satisfies the thin-shell and zero-overlap conditions (C.1) and (C.2), then

$\mathcal{N}(0,\sigma^2)$33

as $\mathcal{N}(0,\sigma^2)$34, and the limit does not depend on the realized $\mathcal{N}(0,\sigma^2)$35 (Bagyan et al., 14 Oct 2025). More generally, if $\mathcal{N}(0,\sigma^2)$36 and $\mathcal{N}(0,\sigma^2)$37 are i.i.d. copies in the Gaussian-modulator setting and

$\mathcal{N}(0,\sigma^2)$38

then the $\mathcal{N}(0,\sigma^2)$39 matrix $\mathcal{N}(0,\sigma^2)$40 satisfies

$\mathcal{N}(0,\sigma^2)$41

where $\mathcal{N}(0,\sigma^2)$42 has independent entries $\mathcal{N}(0,\sigma^2)$43; equivalently,

$\mathcal{N}(0,\sigma^2)$44

that is, a matrix normal limit $\mathcal{N}(0,\sigma^2)$45 (Bagyan et al., 14 Oct 2025).

When $\mathcal{N}(0,\sigma^2)$46 is only spherically symmetric, Schoenberg’s characterization yields a Gaussian scale-mixture representation at the characteristic-function level:

$\mathcal{N}(0,\sigma^2)$47

for some distribution function $\mathcal{N}(0,\sigma^2)$48 on $\mathcal{N}(0,\sigma^2)$49 (Bagyan et al., 14 Oct 2025). This leads to pointwise and uniform convergence of conditional densities and distribution functions to mixtures of centered normal laws $\mathcal{N}(0,\sigma^2)$50 under assumptions (C.1)–(C.3) and integrability conditions (Bagyan et al., 14 Oct 2025).

The quantitative rate is governed by the Gram-matrix deviation term

$\mathcal{N}(0,\sigma^2)$51

which separates thin-shell variance from zero-overlap concentration (Bagyan et al., 14 Oct 2025). This suggests that the asymptotic normality of random isotropic projections is controlled by the extent to which the sample cloud becomes radially concentrated and mutually orthogonal in high dimension.

6. Characterization results, approximations, and statistical inference

A sharp characterization of Gaussian modulators is obtained through Pólya’s theorem. Suppose $\mathcal{N}(0,\sigma^2)$52 satisfies (C.1)–(C.2), is independent of $\mathcal{N}(0,\sigma^2)$53, and $\mathcal{N}(0,\sigma^2)$54 is spherically symmetric with characteristic function $\mathcal{N}(0,\sigma^2)$55. Then

$\mathcal{N}(0,\sigma^2)$56

for some $\mathcal{N}(0,\sigma^2)$57 if and only if, for all $\mathcal{N}(0,\sigma^2)$58,

$\mathcal{N}(0,\sigma^2)$59

as $\mathcal{N}(0,\sigma^2)$60 (Bagyan et al., 14 Oct 2025). Equivalently,

$\mathcal{N}(0,\sigma^2)$61

which is Pólya’s characterization and forces normality (Bagyan et al., 14 Oct 2025). The paper also gives a counterexample using a spherically symmetric $\mathcal{N}(0,\sigma^2)$62-stable modulator, for which the variance condition fails (Bagyan et al., 14 Oct 2025).

For the circle-valued projected isotropic normal, the exact sampling distribution of the mean resultant is analytically intricate, so approximations are built from the von Mises resultant law (Mardia et al., 4 Mar 2026). Two mappings $\mathcal{N}(0,\sigma^2)$63 are proposed. The first is moment matching:

$\mathcal{N}(0,\sigma^2)$64

so $\mathcal{N}(0,\sigma^2)$65 (Mardia et al., 4 Mar 2026). The second is score matching:

$\mathcal{N}(0,\sigma^2)$66

(Mardia et al., 4 Mar 2026). Both satisfy $\mathcal{N}(0,\sigma^2)$67 for small $\mathcal{N}(0,\sigma^2)$68 and $\mathcal{N}(0,\sigma^2)$69 for large $\mathcal{N}(0,\sigma^2)$70 (Mardia et al., 4 Mar 2026).

Inference for the circular model is based on the likelihood

$\mathcal{N}(0,\sigma^2)$71

with numerical maximization (Mardia et al., 4 Mar 2026). Moment-based estimation sets $\mathcal{N}(0,\sigma^2)$72 equal to the sample mean direction and solves

$\mathcal{N}(0,\sigma^2)$73

for $\mathcal{N}(0,\sigma^2)$74 (Mardia et al., 4 Mar 2026). Under uniformity, the Rayleigh statistic satisfies

$\mathcal{N}(0,\sigma^2)$75

for large $\mathcal{N}(0,\sigma^2)$76 (Mardia et al., 4 Mar 2026).

On $\mathcal{N}(0,\sigma^2)$77, parameter estimation proceeds geometrically: compute the sample Fréchet mean $\mathcal{N}(0,\sigma^2)$78, estimate the intrinsic covariance $\mathcal{N}(0,\sigma^2)$79, set $\mathcal{N}(0,\sigma^2)$80, and invert the bijection $\mathcal{N}(0,\sigma^2)$81 numerically to recover $\mathcal{N}(0,\sigma^2)$82 (Figueras et al., 2024). Because only $\mathcal{N}(0,\sigma^2)$83 and $\mathcal{N}(0,\sigma^2)$84 are identifiable from spherical data alone, this is a scale-free estimation procedure unless external information fixes either $\mathcal{N}(0,\sigma^2)$85 or $\mathcal{N}(0,\sigma^2)$86 (Figueras et al., 2024).

For the general sphere-valued projected normal, moment approximations based on Taylor expansions and quadratic-form identities yield analytic approximations to $\mathcal{N}(0,\sigma^2)$87 and $\mathcal{N}(0,\sigma^2)$88 for $\mathcal{N}(0,\sigma^2)$89 with $\mathcal{N}(0,\sigma^2)$90 (Herrera-Esposito et al., 20 Jun 2025). The same paper develops moment-matching estimation using the objective

$\mathcal{N}(0,\sigma^2)$91

with manifold constraints on $\mathcal{N}(0,\sigma^2)$92 and $\mathcal{N}(0,\sigma^2)$93 (Herrera-Esposito et al., 20 Jun 2025). In the isotropic case, these methods can be constrained to $\mathcal{N}(0,\sigma^2)$94 (Herrera-Esposito et al., 20 Jun 2025).

7. Related families, applications, and extensions

The projected isotropic normal appears in several distinct application domains. In neuroscience, a recent phase-analysis model for EEG under flash stimulation assumes

$\mathcal{N}(0,\sigma^2)$95

where $\mathcal{N}(0,\sigma^2)$96 are independent Gaussian with equal variance. The resulting phase $\mathcal{N}(0,\sigma^2)$97 has the projected isotropic normal distribution on the circle (Mardia et al., 4 Mar 2026). The paper studies the mean resultant, the component synchrony measure $\mathcal{N}(0,\sigma^2)$98, approximation-based tests of phase locking, and an EEG application in which electrode O1 under 6 Hz flashing showed $\mathcal{N}(0,\sigma^2)$99 and a hybrid estimate $k$00 (Mardia et al., 4 Mar 2026).

In multivariate visualization, an isotropic reference normal $k$01 projects to circular contours in every 2D view:

$k$02

where $k$03 is the original $k$04-dimensional Mahalanobis threshold (Calvi et al., 4 Feb 2025). This geometry underlies a projection-pursuit index for anomaly detection relative to a multivariate normal baseline, implemented in the R package tourr (Calvi et al., 4 Feb 2025).

In directional and toroidal statistics, isotropic projected normal structure also underlies symmetric, unimodal models on the circle and hypertorus. For the univariate circle case, if $k$05 with $k$06, the projected normal is symmetric around its mean direction $k$07 and its concentration depends only on $k$08 (Kato et al., 22 Aug 2025). The corresponding toroidal projected normal family $k$09 is closed under marginalization, and each univariate marginal is a symmetric, unimodal projected normal on the circle (Kato et al., 22 Aug 2025). This family supports Bayesian inference via latent radii and Metropolis–Hastings or Gibbs-type updates (Kato et al., 22 Aug 2025).

A broader poly-cylindrical extension combines projected normals for circular components with skew-normal linear components, yielding the joint projected normal and skew-normal distribution (Mastrantonio, 2017). That model preserves closure under marginalization, highlights the inherited scale non-identifiability of projected normals, and resolves it by fixing one variance per circular pair and post-processing posterior draws (Mastrantonio, 2017). The isotropic circular case appears as the specialization $k$10 within this larger construction (Mastrantonio, 2017).

Finally, recent generalizations replace normalization by $k$11 with

$k$12

leading to projected distributions on ellipsoids or inside ellipsoids (Herrera-Esposito et al., 20 Jun 2025). These include $k$13 on the unit ball and $k$14 on an ellipsoid interior, with moment approximations and density formulas obtained by linearization and change of variables (Herrera-Esposito et al., 20 Jun 2025). This suggests that the projected isotropic normal is best viewed not as an isolated model, but as the isotropic core of a larger class of normalized Gaussian constructions spanning Euclidean projection, directional statistics, random modulation, and manifold-valued inference.