# High dimensional statistics

March 12, 2015 — October 28, 2021

Placeholder to think about the many weird problems arising in very high dimensional statistical inference. There are many approaches to this problem: throwing out dimensions/predictors as in model selection, considering low dimensional projections, viewing objects with matrix structure for concentration or factorisation, or tensor structure even.

## 1 Soap bubbles

High dimensional distributions are extremely odd, and concentrate in weird ways. For example, for some natural definitions of *typical*, typical items are *not* average items in high See Sander Dielemann’s musings on typicality for an introduction to this plus some motivating examples.

For another example, consider this summary result of Vershynin (2015):

Let \(K\) be an isotropic convex body (e.g. an \(L_2\) ball) in \(\mathbb{R}^{n},\) and let \(X\) be a random vector uniformly distributed in \(K\), with \(\mathbb{E}X=0\) and \(\mathbb{E}XX^{\top}=I_n.\) Then the following is true for some positive constants \(C,c\):

- (Concentration of volume) For every \(t \geq 1\), one has \[ \mathbb{P}\left\{\|X\|_{2}>t \sqrt{n}\right\} \leq \exp (-c t \sqrt{n}) \]
- (Thin shell) For every \(\varepsilon \in(0,1),\) one has \[ \mathbb{P}\left\{\left|\|X\|_{2}-\sqrt{n}\right|>\varepsilon \sqrt{n}\right\} \leq C \exp \left(-c \varepsilon^{3} n^{1 / 2}\right) \]

That is, even with the mass *uniformly distributed* over space, as the dimension grows, it all ends up in a thin shell, because volume grows exponentially in dimension. This is popularly known as a soap bubble phenomenon. This is one of the phenomena that leads to interesting behaviour in low dimensional projection. The more formal name is the *Gaussian Annulus Theorem*. Turning it around, for a d-dimensional spherical Gaussian with unit variance in each direction, for any \(\beta \leq \sqrt{d}\), all but at most \(3 e^{-c \beta^{2}}\) of the probability mass lies within the annulus \(\sqrt{d}-\beta \leq|\mathbf{x}| \leq \sqrt{d}+\beta,\) where \(c\) is a fixed positive constant.

## 2 Convex hulls

Balestriero, Pesenti, and LeCun (2021) cite Bárány and Füredi (1988):

Given a \(d\)-dimensional dataset \(\boldsymbol{X} \triangleq\left\{\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{N}\right\}\) with i.i.d. samples \(\boldsymbol{x}_{n} \sim \mathcal{N}\left(0, I_{d}\right), \forall n\), the probability that a new sample \(\boldsymbol{x} \sim \mathcal{N}\left(0, I_{d}\right)\) is in interpolation regime (recall Def. 1 ) has the following limiting behavior \[ \lim _{d \rightarrow \infty} p(\underbrace{\boldsymbol{x} \in \operatorname{Hull}(\boldsymbol{X})}_{\text {interpolation }})= \begin{cases}1 & \Longleftrightarrow N>d^{-1} 2^{d / 2} \\ 0 & \Longleftrightarrow N<d^{-1} 2^{d / 2}\end{cases} \]

They observe that this implies high dimensional statistics rarely interpolates between data points, which is not surprising, but only in retrospect. Despite some expertise in high-dimensional problems I had never noticed this fact myself. Interestingly they collect evidence that suggests that low-d projections and latent spaces are *also* rarely interpolating.

## 3 Empirical processes in high dimensions

Combining empirical process theory with high dimensional statistics gets us to some interesting models. See, e.g. van de Geer (2014b).

## 4 Markov Chain Monte Carlo in high dimensions

TBD

## 5 References

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*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.