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Consider the following process:

• Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \in \mathbb R_{\ge 0} \}$. Project $S$ to this cone.

What is the minimum dot product between two unit vectors in $S$, after $L$ iterations? Can it be bounded away from zero?

See the illustration of the process for $n = 2$. The red area outlines the set $S$. After projecting to the cones defined by the blue, green, and orange bases, the set $S$ becomes contained in a line.

The regime of interest is ideally $L = O(1)$, but $L = o(n)$ could also work.

It may be that the following stepping stone is more tractable:

• Start with a set $S = \mathbb R^n$. Repeat $nL$ times: choose a random unit vector $u$, and consider the halfspace $H = \{ x \in \mathbb R^n : u^Tx \ge 0 \}$. Project $S$ to this halfspace.

Consider the following process:

• Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \in \mathbb R_{\ge 0} \}$. Project $S$ to this cone.

What is the minimum dot product between two unit vectors in $S$, after $L$ iterations? Can it be bounded away from zero, say with high probability in $n$?

See the illustration of the process for $n = 2$. The red area outlines the set $S$. After projecting to the cones defined by the blue, green, and orange bases, the set $S$ becomes contained in a line.

The regime of interest is ideally $L = O(1)$, but $L = o(n)$ could also work.

Consider the following process:

• Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \in \mathbb R_{\ge 0} \}$. Project $S$ to this cone.

What is the minimum dot product between two unit vectors in $S$, after $L$ iterations? Can it be bounded away from zero?

See the illustration of the process for $n = 2$. The red area outlines the set $S$. After projecting to the cones defined by the blue, green, and orange bases, the set $S$ becomes contained in a line.

The regime of interest is ideally $L = O(1)$, but $L = o(n)$ could also work.

It may be that the following stepping stone is more tractable:

• Start with a set $S = \mathbb R^n$. Repeat $nL$ times: choose a random unit vector $u$, and consider the halfspace $H = \{ x \in \mathbb R^n : u^Tx \ge 0 \}$. Project $S$ to this halfspace.

We can prove that $S$ will become measure zero in both processes; however, it doesn't imply the sort of concentration of angle in the question.

Motivation: this is a toy version of a technical result concerning random neural networks that came up in my research. Any properties of this set $S$, for example its Gaussian width, are welcome.

Consider the following process:

• Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \in \mathbb R_{\ge 0} \}$. Project $S$ to this cone.

What is the minimum dot product between two unit vectors in $S$, after $L$ iterations? Can it be bounded away from zero?

See the illustration of the process for $n = 2$. The red area outlines the set $S$. After projecting to the cones defined by the blue, green, and orange bases, the set $S$ becomes contained in a line.

The regime of interest is ideally $L = O(1)$, but $L = o(n)$ could also work.

It may be that the following stepping stone is more tractable:

• Start with a set $S = \mathbb R^n$. Repeat $nL$ times: choose a random unit vector $u$, and consider the halfspace $H = \{ x \in \mathbb R^n : u^Tx \ge 0 \}$. Project $S$ to this halfspace.

Consider the following process:

• Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \in \mathbb R_{\ge 0} \}$. Project $S$ to this cone.

What is the maximum dot product between two unit vectors in $S$, after $L$ iterations? Can it be bounded away from zero?

See the illustration of the process for $n = 2$. The red area outlines the set $S$. After projecting to the cones defined by the blue, green, and orange bases, the set $S$ becomes contained in a line.

The regime of interest is ideally $L = O(1)$, but $L = o(n)$ could also work.

It may be that the following stepping stone is more tractable:

• Start with a set $S = \mathbb R^n$. Repeat $nL$ times: choose a random unit vector $u$, and consider the halfspace $H = \{ x \in \mathbb R^n : u^Tx \ge 0 \}$. Project $S$ to this halfspace.

We can prove that $S$ will become measure zero in both processes; however, it doesn't imply the sort of concentration of angle in the question.

Motivation: this is a toy version of a technical result concerning random neural networks that came up in my research. Any properties of this set $S$, for example its Gaussian width, are welcome.

Consider the following process:

• Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \in \mathbb R_{\ge 0} \}$. Project $S$ to this cone.

What is the minimum dot product between two unit vectors in $S$, after $L$ iterations? Can it be bounded away from zero?

See the illustration of the process for $n = 2$. The red area outlines the set $S$. After projecting to the cones defined by the blue, green, and orange bases, the set $S$ becomes contained in a line.

The regime of interest is ideally $L = O(1)$, but $L = o(n)$ could also work.

It may be that the following stepping stone is more tractable:

• Start with a set $S = \mathbb R^n$. Repeat $nL$ times: choose a random unit vector $u$, and consider the halfspace $H = \{ x \in \mathbb R^n : u^Tx \ge 0 \}$. Project $S$ to this halfspace.

We can prove that $S$ will become measure zero in both processes; however, it doesn't imply the sort of concentration of angle in the question.

Motivation: this is a toy version of a technical result concerning random neural networks that came up in my research. Any properties of this set $S$, for example its Gaussian width, are welcome.