Split (1)

train · 282k rows

text string	source string
●●● ● ●●● ● ● ●●● ● ●● ●● ●● ●● ●● ● ●●● ●●● ●● ●●●● ●● ●●● ●●●●● ●●● ●●●● ● ●●●● ●●● ●●●● ●● ● ●● ● ● ●● ●●● ● ●● ●● ●● ●● ●●● ● ● ●● ● ●● ●● ●●● ●● ●●●● ●●●● ●● ●●● ● ●●●● ●● ●● ● ●●●● ● ●● ● ● ●●●● ●●● ● ●●● ●●●● ●● ●●● ● ● ●●●● ● ●● ●● ●● ●● ●● ●●● ●●● ●● ● ●● ●● ●● ●● ● ● ●● ● ●●●● ●● ● ●● ● ●● ●● ●●● ● ● ● ● ●●● ●●● ●●●●●● ●●● ● ●● ● ●●● ●● ●● ●● ● ● ●●● ● ●●● ●●● ●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●● ●● ●●● ● ●●● ● ●●●●● ● ●● ● ●● ●● ●● ● ● ●●●●●●● ● ●● ● ●● ● ●● ●●● ●●● ●● ● ●● ● ●● ● ●●●● ●● ●● ●●● ●●● ●●● ●● ●● ●●● ●● ●● ●● ● ●●●● ● ● ●●●● ●● ● ●●● ●● ●● ●●● ● ●●● ●●●● ●● ●●● ●●●● ●● ● ●● ●● ● ●●● ● ●●● ●● ● ●●● ●●●● ● ●●●● ●●● ●● ●● ● ●● ● ●● ●● ●● ● ●● ● ● ●● ●● ●● ● ●● ●● ●● ● ●● ● ●● ● ●● ● ● ●●● ● ●●●● ●● ●● ● ● ●●● ● ●●● ●● ●● ● ● ●● ● ●●●●● ● ●● ●●●● ● ●● ●● ●● ●● ●● ●● ●●● ●●●●●● ● ●●●● ●●●● ● ●● ●●● ●● ● ●● ●● ●● ●●● ●● ●●● ● ●●● ●●● ● ● ●● ● ● ●●●● ●● ●● ● ●●● ●● ●● ● ●●● ●● ●● ●● ● ●● ●●●● ●● ● ●●● ●● ●●● ●●● ●● ●●● ●●●● ● ●●● ●● ●● ●● ●●● ●● ●● ●●●●● ●● ●●●● ●● ● ●●● ●●●● ●●●●● ●● ●● ●● ●●●● ● ●●● ● ● ●●● ● ●●●● ●● ●● ●● ●● ●● ●● ● ●●● ●●● ● ●● ●● ●●●● ●●● ● ●● ●● ●●● ● ●● ●● ● ●● ● ●●●●●● ● ●●● ● ●●●●● ● ●● ●● ●●● ● ●● ●● ●● ●● ●● ● ●●● ●●● ●● ●● ●●● ●● ●● ●●●● ●● ●●●●●● ●● ● ●●●● ●●● ●● ● ●●● ● ●● ● ● ●●● ●● ●● ●●● ●● ●● ● ●● ● ● ●●●● ●● ●● ●● ● ●●● ●●● ●● ●● ●● ● ●● ●● ●●● ● ● ●●● ●● ●● ●● ●● ●● ●●● ● ●● ● ●●● ●● ● ● ●● ● ●●● ●●● ●●●● ● ●● ●● ●●●● ●●● ● ●● ● ●● ● ●●●● ●● ●● ●● ●●● ● ●● ● ●● ●● ● ●● ●● ●● ●● ●●● ● ●●●● ● ● ●● ●● ●● ●●●● ●●● ●●● ●● ●● ●●● ● ● ●● ●● ● ●● ● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●●● ● ●●●● ●● ●● ●●● ●●● ●●● ●● ●●●● ●●●● ●●● ●●● ●● ● ●● ● ●●●● ●●● ●●●●● ● ●●	https://arxiv.org/abs/2504.19138v1
● ●● ● ●● ●●●● ●● ●● ●●● ●● ●● ● ●● ●● ●● ●●● ●●●●● ●●●● ● ●● ●● ●●● ●● ●●●●● ● ●● ●● ●● ●● ●●● ●● ● ●● ●●● ● ●● ●● ●● ●● ●●●● ● ●●● ● ●● ●● ●●●● ●●● ●● ●● ●● ●● ● ●●●● ● ●●● ● ●●● ●● ●●● ●●●● ●● ●●●● ● ●●● ●● ● ● ●●● ● ●● ●●● ●●● ● ●●●● ● ●● ● ●● ● ● ●● ●● ●● ●●● ● ●● ● ●● ● ●● ●●● ● ●● ● ● ●●● ●● ●● ● ● ● ●● ●●●● ● ●●●●● ●● ● ● ●●● ● ●●●● ●● ●●● ● ●● ●●● ●● ●● ● ●● ●●●● ●● ●● ●●● ●● ● ●● ●●● ●● ●● ● ●● ●● ●●● ●●● ●●● ●● ●● ●● ●● ●●● ●● ●●●● ●●● ●●● ● ●● ● ●●● ● ● ●●● ●● ●●●● ●●● ●● ●●● ●●● ●●●● ●●●● ●● ● ●● ● ● ●● ● ●● ● ●●●● ●●● ● ● ●● ● ●● ● ●●●● ● ●●●● ●●● ● ●● ●●● ●●●● ●●●● ●● ●● ● ●● ●●● ● ● ●●● ●● ●●●● ●● ●● ● ● ●● ●● ●● ●● ● ●●●● ●●● ●●● ●● ● ●● ●● ●● ●●●●● ●●● ●●● ●● ● ●●● ● ●●● ●●● ●●● ●● ●● ● ● ●● ●● ●● ●●● ● ●●● ●● ● ● ●● ●●●● ●●● ●●● ●●● ● ●● ●●●●● ● ●●● ●● ●● ●●● ●● ● ●● ● ●● ●●●● ● ●●● ●● ●● ●● ●● ●●●● ● ●●●● ● ●● ●● ●●●●● ● ●● ●●● ● ●● ●● ● ●● ● ●● ●● ●● ● ●●●● ● ●● ● ●● ●●● ●● ●●●● ● ●●● ● ●●●● ● ● ●●● ● ● ●●●●●● ● ●●●● ●● ●●● ●● ●● ● ●●● ●● ●● ●●●● ● ●● ●● ●●● ●● ● ●● ●● ●●● ●● ●●●●●●● ●● ● ●●● ●● ●● ● ●● ●●● ●● ●●●●● ●●●●● ●● ●●● ● ●●● ●●● ●●● ●●● ● ●●● ● ●●● ● ● ●●●● ● ●●●● ●●●● ●● ●●●● ● ● ●●● ●● ●●● ●● ●● ●●●●● ●●● ●● ● ●● ●● ●● ● ● ●● ●● ● ●●● ●●●● ● ●●● ● ●● ●● ●● ●● ●● ● ●● ●●● ●● ●●● ●●● ● ●●● ● ●●●● ● ●● ● ●● ●●● ● ● ●●● ●●● ●●● ● ●● ● ●● ● ●● ●● ● ●●● ●●● ●● ●●● ●● ●●● ●● ●●● ● ●● ●●● ●● ●● ●●● ●● ●● ●● ● ●● ●● ●● ●●● ●●● ●●● ●●●● ● ● ●●●●●● ● ● ●●● ●●● ● ●●● ●● ●●● ●● ●●● ●● ●● ●● ● ●● ● ●●●● ● ●●● ● ●● ●●● ● ●● ●●●● ●●● ●●● ● ● ●●●● ●● ●● ●●● ●●● ● ●●● ●●● ● ●●● ●●●● ●● ●●● ●●● ●● ●● ● ●● ●●● ●●● ●●● ●●● ●● ● ●● ●● ● ● ●● ● ●●● ● ●●● ●● ● ●●●●● ● ●●● ●● ●● ●● ● ● ● ●● ●● ● ●●	https://arxiv.org/abs/2504.19138v1
●●● ● ●●● ● ● ●● ● ●● ● ●● ●● ●●● ● ●●● ● ●● ●● ● ●●●●● ●● ●●●● ●● ● ●● ●●●● ●●●● ●● ● ●● ● ●● ●●● ●● ●● ● ●●● ●●● ●●● ● ● ●●● ●● ●● ●● ●● ●● ●●●●● ●● ●●● ● ●●● ● ●●●●● ● ●●●● ● ●● ● ●●● ●● ●● ● ●●● ● ●●●● ●●● ●●● ● ●●● ●● ●●● ●● ●● ●● ●●● ● ●●●● ●● ● ●● ●●● ● ●● ●● ●●●● ●● ●●● ● ●● ●●● ●●● ●●●● ●●● ● ●● ●● ●● ● ● ●● ●●● ● ●●● ●● ●●● ●●●●● ● ●● ● ●●●● ●●● ●● ●●●● ●● ●● ● ●● ●●● ●● ● ●● ●● ●● ●● ●●●●●● ●●●● ● ● ●● ●● ●●● ●● ● ●● ●● ●● ●● ●●●●●● ●● ●●● ●● ●● ●● ●● ● ●● ●● ● ● ●● ●●●● ●● ●● ●●● ● ●●● ●●●● ● ●● ●● ● ●●●● ●● ●● ●● ●●●●● ● ●● ● ●● ●● ●● ●● ●●● ●● ● ●●●●●● ●●●●●● ●●● ●●●●●● ●● ●● ●● ●● ●●● ●●●●● ● ●●● ●● ●● ● ● ● ●●● ●● ●●● ●●● ● ●● ●●● ●● ●● ●● ●●● ●● ● ●●●●● ●● ●● ● ●● ●● ●● ●●●●●●● ●● ● ● ●●● ●● ●●● ● ● ●●●● ●●●● ● ●●●●●● ● ●● ● ●● ●●● ●● ●● ●● ●● ● ●●●●●●● ● ● ●●● ●●● ● ●● ●●● ●● ●●● ●●●● ● ● ● ●● ●●● ● ● ●●●●● ●●● ●● ●●● ●●● ●● ●●● ● ●● ● ●● ●●● ●●● ●● ● ●●● ●●● ●● ● ● ●●●● ●● ●● ●● ●●●●● ●● ●● ●● ● ●●● ● ●● ●●● ● ●●●●● ● ●●● ●● ●●● ●● ●●● ●●● ● ●●● ●●● ●●● ●● ● ●● ●●●● ● ●●●●●●● ●●● ●●● ●● ●● ●●● ● ● ●● ●●●● ● ●● ●●● ● ●● ●●●●● ●● ●●●● ●●● ●●● ●● ●● ●●●● ●● ●●●● ●● ●● ●● ●● ● ●●●●● ●● ●● ● ●●●●● ● ●●● ●● ● ●● ●●●● ● ●●● ●● ●●●● ●●● ● ●● ● ●●●●● ●● ●● ●●● ● ●● ● ●●● ●● ●● ●● ●● ●● ●●● ●● ● ●● ●●●●● ●● ●● ●●● ● ●● ●● ● ●●● ● ●●●● ● ●● ● ●● ●● ●●● ●●● ● ●●●● ●● ● ● ●●●● ● ●●● ● ●●● ●● ●●● ●●● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●●●● ●● ●●● ●●● ●●● ●● ●● ●●● ●● ●● ● ●●● ● ●●● ●● ●●● ● ●● ●● ●● ●● ●●●●● ●●● ●●● ●● ●●● ● ●●● ● ●● ●● ●● ●● ●●● ●● ●● ●● ●● ● ●● ● ●● ●● ●● ●● ● ●● ●●●● ●● ●● ●●●● ●● ●●●●● ● ● ●●●● ●●●● ●●● ●● ● ●● ●●● ●●●● ●● ● ●● ●●● ●●●● ●● ● ●● ● ●●● ●● ●● ●● ● ●●● ● ●● ●●● ●●● ●● ● ●● ●● ●●● ●●● ● ●●●● ●● ●● ●●● ●● ●●● ●● ●● ●●● ● ●● ●	https://arxiv.org/abs/2504.19138v1
●● ●● ●●● ●● ●● ● ● ●●● ● ●● ● ●●● ●● ● ● ●● ●●● ●● ●●● ● ●● ●● ●●● ● ●● ●● ●●● ● ● ●●●● ●● ●● ● ●● ●●● ●● ● ● ●●●●● ● ●● ●● ●● ●● ●● ●●● ●● ●●● ● ●● ●●● ●● ●●●● ● ●●● ●●●● ●● ●● ●● ● ●●●● ● ●● ●●● ● ●● ●●● ● ●●● ● ●● ● ●● ● ●●● ● ● ●●● ●●●●●● ●● ●● ●● ●●● ● ●●● ●● ●●● ● ●●●● ● ●● ●● ●●● ●●● ● ●●● ●●●●●● ●●● ●● ●●●● ●●●● ●● ●●●● ●●● ●● ● ●● ●●● ●● ●●● ●●●● ● ●●● ●● ●●● ●● ● ● ● ● ●● ● ● ● ●●● ●●●● ●● ● ●● ●●● ● ●●●● ● ●●● ● ●● ●● ●●●● ●● ●●●●● ●●● ● ●● ●● ● ● ●● ●●●● ● ●●● ●●●● ●● ●● ●● ● ●●●●● ● ●● ●●● ●● ●● ●● ●● ●● ● ●● ●●● ●●● ●●●● ● ●●● ● ● ●● ● ●● ●●● ● ●● ● ●●●● ●●● ●● ●● ●● ●● ●● ●● ●●● ● ●● ● ●● ●● ● ● ● ●●● ●● ●● ●●●● ●●●● ●●● ●●● ●●● ●● ●● ● ●●●●● ● ● ●● ●● ●● ●●● ●● ● ●● ●●●●●● ●● ●●● ●● ●●● ● ● ● ●●● ●●●● ●● ●●●● ●●●● ● ●●● ●● ● ●● ● ●● ● ● ●● ● ●●● ● ●●● ●● ● ●●● ● ● ●●● ●● ●●●● ●● ●●● ●● ● ●●● ●● ●●●● ● ●● ●● ●● ● ●● ● ● ●●●● ● ● ●● ●●●●● ●● ●●● ●● ● ●● ●●● ●● ●● ●●●●● ●● ● ●●● ●●● ●●● ●● ●● ●● ●● ●● ●●● ●● ●● ●●● ●●● ●● ●●●● ● ● ●●●● ●● ●● ●● ● ●● ● ●● ●●●● ●●● ●● ●● ●● ● ●●● ●●● ● ●●● ● ●● ●● ●●● ●●●● ●●● ●●● ●● ● ●● ●● ● ●● ● ●● ●● ● ● ●● ●● ● ● ● ●●● ●● ●● ● ●●● ●● ●●● ● ●● ●● ●●●● ●●●● ●● ●● ● ●●● ●● ●●● ●● ● ●● ●● ●● ●● ● ●● ●●●● ● ●● ●●● ●● ● ●● ●●● ●● ●●● ●● ● ●● ● ●●● ●●●● ●●● ● ●●●● ● ●● ●●●● ● ● ●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●● ●●● ●● ●●● ● ●●● ●● ●● ●●●●● ●● ●●● ● ●● ●●●● ●●● ●●● ●●● ●●● ●● ●●● ● ●●● ●● ●● ●● ●● ● ●●● ● ●● ●● ●●● ● ●● ●● ●● ●●●●●●● ●●●● ●● ●● ●● ● ●● ●●● ●●●● ●● ●● ● ●● ●●●● ● ●●● ●● ●● ●● ●●● ● ●● ●●●● ● ●●● ●● ●● ●● ●● ●● ● ● ●●●● ●● ●●● ●●● ●●●● ●●● ●● ●●● ●● ●● ●● ●●●● ●● ● ● ●●● ● ● ●● ● ● ●● ● ●●● ●● ●● ●● ●● ●● ●● ●●●●● ●●● ●●● ● ●● ●●● ● ●● ● ●●● ●●● ●●	https://arxiv.org/abs/2504.19138v1
●● ● ● ●●● ● ●●● ●●● ● ●●●●● ●●● ●●● ●● ● ●● ●●● ●● ● ●●● ● ●●● ●● ● ●● ●●● ● ● ●●●● ●● ● ● ●●● ●●● ●●● ●● ● ● ●● ● ●● ●●● ●●●●● ●● ●● ●● ● ● ●● ● ●● ●●●● ● ● ●● ●●●●● ●● ●● ●●●● ●●● ●● ●● ●● ●●●● ● ●● ●● ●● ●● ●●● ●● ●●● ●●●●● ● ●● ●●●● ●● ● ●● ● ●●● ●●● ●●● ●●● ●● ● ●● ●●●● ●● ●● ●●●● ●●●● ● ●●● ●● ●● ●● ●●● ●●● ●● ● ●● ●●● ●●● ● ●● ● ● ●● ●●●●● ●●● ● ●●● ●● ●●●● ●●● ●● ● ● ●●●● ●●● ●● ● ● ●●● ●● ● ●●● ● ●● ● ●● ● ●●● ●● ●● ●● ●●● ● ●●●● ●● ●● ●● ● ●● ● ●●●●●● ●●● ● ●● ●●●● ●●●●● ●●●●● ●● ● ●● ● ●●● ●● ●●● ●●●● ●●●● ●● ●●● ●●● ● ● ●●●●●● ● ● ● ●●● ●● ●●●● ●● ●● ●● ●● ●● ●●● ● ●● ●● ●●●● ● ● ●● ●●● ●●● ● ●●● ● ●● ●● ● ● ●● ●●●● ● ●●●●● ● ●● ●●●● ●●●● ●●● ●●● ●●●● ● ●● ●●●● ●● ●● ●●● ●● ● ●● ●● ● ●● ●●● ●●● ●● ● ●● ● ●● ●● ●●● ● ●● ● ●● ●● ●● ●● ●●●● ●●●●● ● ●● ●●●● ● ●● ●●● ●●● ●●●●● ● ●● ●●● ●●● ●●●●● ●● ●● ●●● ● ●●●● ●● ●●●● ●●● ●● ●● ●● ● ●● ● ●●● ●●● ●●● ●● ● ●●● ●● ● ●●● ●● ●●● ● ●●● ●●● ● ●●● ● ●● ● ●● ● ●● ●● ●● ●● ●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●● ●● ●● ● ●●● ●● ● ●●●●●● ●● ●●● ● ●● ●● ●●● ●● ● ●● ● ●● ●● ●● ●●● ● ●● ● ●●● ● ●●●● ●● ●●● ●● ●●● ●●● ● ●●●●● ● ●●● ● ●●● ● ●● ●●●● ●●● ● ●● ●● ● ●●● ●●● ●●● ●● ●● ●● ●●● ●● ●● ●● ●●● ●● ●● ●●● ●● ●● ●●●●●● ●●● ●● ●● ● ●●● ●● ● ● ●●● ●●● ●● ●●● ●● ●●● ● ● ●●●● ●●●● ●● ● ●●●●● ●●●●● ●●●●●● ●●● ●● ●●●● ●●●● ●●●●● ● ● ●● ● ●● ● ●● ●● ●● ●● ● ●● ●●● ● ● ●● ●● ●●● ●●● ●●● ●●● ●●● ●●●● ●●● ●●●● ●●● ●●● ●●● ●●● ● ●● ●●●●● ● ●● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ● ●●●●●● ●●●● ●● ●●●● ●● ●●● ●●●●●● ● ●● ●●● ● ● ●● ●● ●● ● ●●● ●●●● ● ●● ● ●●●● ● ●●● ●● ●● ●● ● ●●●● ● ●● ●● ●● ●● ●● ● ●●● ●● ●● ●● ●●● ● ● ●● ●● ●● ● ●● ● ●● ●●● ● ●● ●●● ●●● ● ●●● ●● ●●● ● ●● ● ●●●● ●● ●● ●● ●●● ●●●●●● ● ● ● ●● ● ●● ●●●● ●● ●●●● ●● ●● ●●	https://arxiv.org/abs/2504.19138v1
● ● ●●● ●● ●● ●● ●● ●● ●●● ● ●● ● ●● ● ●●● ●● ●●● ●● ●● ● ●●●● ●● ●● ● ●● ●● ●●● ●●●● ● ●●● ●●● ● ●●● ● ●● ●● ●● ●●● ●●● ● ●●●● ●●● ●● ●● ●●●● ● ●●● ●● ●● −4 −2 0 2 4−0.006 −0.004 −0.002 0.000 0.002 0.004 0.006Normal Q−Q Plot Theoretical QuantilesSample Quantiles Figure 7: Histogram and normal quantile-quantile plot of ˆ µE−µunder RLS when m= 18. 35 Pr(ˆµE> µ) to 1 /2 is markedly slower than in the one-dimensional case, with RLS and CRD exhibiting comparable rates. Figure 5 and 6 compare the 90th percentile interval lengths and empirical coverage levels, respectively. While quantile intervals outperform t-intervals under CRD, the opposite is true under RLS, due to the fact that ˆ µE−µunder RLS is approximately normal for the range of mwe are testing. Although our theory predicts the distribution of ˆµE−µbecomes concentrated and heavy-tailed asymptotically, the curse of di- mensionality delays these effects. At m= 18, RLS errors exhibit only marginally heavier tails than a normal distribution (Figure 7). 6 Discussion Our analysis has so far focused on infinitely differentiable integrands. A main obstacle in extending our results to finitely differentiable integrands is the decay of Walsh coefficients is insufficient to decompose ˆ µ∞−µin a manner compat- ible with Lemma 3. To illustrate this, consider the case where fhas square- integrable dominating mixed derivatives of order 1 ( f\|κ\|(x) for\|κ\| ∈ { 0,1}s). By Corollary 3 of [19] with α= 0, λ= 1, for any ℓ= (ℓ1, . . . , ℓ s)∈Ns ∗, X k∈Bℓ,s\|ˆf(k)\|2⩽Cf,s4−Ps j=1ℓj, where Bℓ,s={k∈Ns ∗\| ⌈κj⌉=ℓjforj∈1:s}andCf,sis a constant depending onfands. Mimicking our proof strategy in Section 3, one might set Km=Q′ Nm with Q′ N=n k∈Ns ∗ k∈Bℓ,sforℓ∈Ns ∗satisfyingsX j=1ℓj⩽No and attempt to tune Nmto satisfy Lemma 3. However, unlike the original QN, the set Q′ Nis rich in additive relations, particularly when s= 1, as Q′ Nforms aF2-vector space. This restricts our choice of Nmand makes the condition limm→∞Pr(\|SUM 1\|⩽\|SUM 2\|) = 0 harder to hold. Thus, our proof strategy cannot be naively applied to finitely differentiable integrands. A second critical limitation is the curse of dimensionality, which our asymp- totic analysis does not fully resolve. While Nm∼λm2/sensures Nm>> m in the limit, practical high-dimensional settings may yield Nm< m. In such cases, SUM 1is close to an empty sum and the bounds in Corollary 2 are non- informative. A finite-sample analysis is therefore required to explain phenomena like the rapid descent in Figure 4 for small m. One promising direction, inspired by Section 6 of [21], is to replace the dimension sin the analysis by a finite- sample effective dimension that captures the integrand’s low-dimensional struc- ture. How to adapt such a framework to our setting is an interesting question for future research. A natural follow-up question concerns the limiting distribution of ˆ µ∞−µ. By Theorem 2 and Corollary 2, we can replace ˆ µ∞−µby SUM′ 1when study its 36 limiting distribution. The difficulty lies in the joint dependencies among Z(k). For	https://arxiv.org/abs/2504.19138v1
large m, we conjecture that SUM′ 1can be approximated by SUM′′ 1=X k∈QNmZ′(k)S′(k)ˆf(k), where each Z′(k) is sampled independently from a Bernoulli distribution with success probability 2−m. This approximation holds rigorously for polynomial in- tegrands, where the support of non-zero Walsh coefficients is particularly sparse. How to extend this result to general integrands is another challenging question for future research. A critical limitation of quantile-based confidence intervals lies in their finite- sample coverage guarantees. When ris odd and ℓ=r−u, the coverage proba- bility of [ˆ µ(ℓ) E,ˆµ(u) E] is structurally bounded above by the nominal level. In appli- cations where undercoverage poses significant risks, the conventional t-interval, despite its slower convergence rate, may remain preferable due to its conservative bias. It remains an open problem how to design intervals that simultaneously achieve adaptive convergence rates and robust finite-sample coverage. Acknowledgments The author acknowledges the support of the Austrian Science Fund (FWF) Project DOI 10.55776/P34808. For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. This paper is developed from a chapter of the author’s thesis. I would like to thank Professor Art Owen for his mentorship and many helpful suggestions. References [1] K. Basu and R. Mukherjee. Asymptotic normality of scrambled geometric net quadrature. The Annals of Statistics , 45(4):1759–1788, 2017. [2] N. Clancy, Y. Ding, C. Hamilton, F. J. Hickernell, and Y. Zhang. The cost of deterministic, adaptive, automatic algorithms: Cones, not balls. Journal of Complexity , 30(1):21–45, 2014. [3] J. Dick. Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. The Annals of Statistics , 39(3):1372–1398, 2011. [4] J. Dick and F. Pillichshammer. Digital sequences, discrepancy and quasi- Monte Carlo integration . Cambridge University Press, Cambridge, 2010. [5] P. Erd¨ os. On a lemma of Littlewood and Offord. Bulletin of the American Mathematical Society , 51(12):898 – 902, 1945. 37 [6] B. Fristedt. The structure of random partitions of large integers. Transac- tions of the American Mathematical Society , 337(2):703–735, 1993. [7] M. Gnewuch, P. Kritzer, A. B. Owen, and Z. Pan. Computable error bounds for quasi-monte carlo using points with non-negative local discrep- ancy. Information and Inference: A Journal of the IMA , 13(3):iaae021, 08 2024. [8] E. Gobet, M. Lerasle, and D. M´ etivier. Mean estimation for randomized quasi monte carlo method. Hal preprint hal-03631879v2 , 2022. [9] A. Griewank, F. Y. Kuo, H. Le¨ ovey, and I. H. Sloan. High dimensional integration of kinks and jumps–Smoothing by preintegration. Journal of Computational and Applied Mathematics , 344:259–274, 2018. [10] S. Joe and F. Y. Kuo. Constructing Sobol’ sequences with better two-dimensional projections. SIAM Journal on Scientific Computing , 30(5):2635–2654, 2008. [11] S. G. Krantz and H. R. Parks. A primer of real analytic functions . Springer Science & Business Media, 2002. [12] P. L’Ecuyer, M. K. Nakayama, A. B. Owen, and B. Tuffin. Confidence intervals for randomized quasi-monte carlo estimators. Technical report, hal-04088085, 2023. [13] S. Liu and A. B. Owen. Preintegration via active subspace.	https://arxiv.org/abs/2504.19138v1
SIAM Journal on Numerical Analysis , 61(2):495–514, 2023. [14] W.-L. Loh. On the asymptotic distribution of scrambled net quadrature. Annals of Statistics , 31(4):1282–1324, 2003. [15] J. Matouˇ sek. On the L2–discrepancy for anchored boxes. Journal of Com- plexity , 14:527–556, 1998. [16] M. K. Nakayama and B. Tuffin. Sufficient conditions for central limit the- orems and confidence intervals for randomized quasi-monte carlo meth- ods. ACM Transactions on Modeling and Computer Simulation , 34(3):1– 38, 2024. [17] A. B. Owen. Randomly permuted ( t, m, s )-nets and ( t, s)-sequences. In H. Niederreiter and P. J.-S. Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing , pages 299–317, New York, 1995. Springer-Verlag. [18] A. B. Owen. Error estimation for quasi-Monte Carlo. Preprint , 2025. [19] Z. Pan. Automatic optimal-rate convergence of randomized nets using median-of-means. Mathematics of Computation , 2025. 38 [20] Z. Pan and A. B. Owen. Skewness of a randomized quasi-monte carlo estimate. Preprint , 2024. [21] Z. Pan and A. B. Owen. Super-polynomial accuracy of multidimensional randomized nets using the median-of-means. Mathematics of Computation , 93(349):2265–2289, 2024. [22] I. M. Sobol’. The distribution of points in a cube and the accurate evalu- ation of integrals (in Russian). Zh. Vychisl. Mat. i Mat. Phys. , 7:784–802, 1967. [23] K. Suzuki and T. Yoshiki. Formulas for the walsh coefficients of smooth functions and their application to bounds on the walsh coefficients. Journal of Approximation Theory , 205:1–24, 2016. [24] J. Wiart, C. Lemieux, and G. Y. Dong. On the dependence structure and quality of scrambled (t, m, s)-nets. Monte Carlo Methods and Applications , 27(1):1–26, 2021. 39 Appendix This appendix contains the proofs of Lemma 5, 7 and 9. The proof strat- egy is inspired by [6]. To simplify the notation, we write aN=O(bN) if lim supN→∞\|aN\|/\|bN\|< C for some constant C > 0 and aN=o(bN) if limN→∞\|aN\|/\|bN\|= 0. We first construct an importance sampling measure on k∈Ns 0. Recall that each k∈N0corresponds to κ⊆Nthrough k=P ℓ∈κ2ℓ−1. Let LN(k) be the likelihood function of kunder the importance sampling measure described by LN(k) =sY j=1∞Y ℓ=1qℓ N 1 +qℓ N1{ℓ∈κj}1 1 +qℓ N1{ℓ/∈κj} =q∥k∥1 NQ∞ ℓ=1(1 +qℓ N)s with qN= exp( −πp s/12N). The value of qNis chosen so that LN(k) closely approximates U(QN) with QNdefined by equation (13). Under LN(k), it is clear that Xjℓ=1{ℓ∈κj}equals 1 with probability qℓ N/(1 + qℓ N) and {Xjℓ, j∈1:s, ℓ∈N}are jointly independent. We use Pr ,E,Var to denote the probability, expectation and variance when kfollows a U(QN) distribution and PrL,EL,VarLto denote those under the importance sampling measure LN(k). Suppose we are interested in Pr( k∈A) =\|A\|/\|QN\|for a subset A⊆QN. We can compute it under the importance sampling measure by Pr(k∈A) =ELh1(k∈A) \|QN\|L(k)i =ELh1(k∈A) \|QN\|q∥k∥1 N∞Y ℓ=1(1 +qℓ N)si . (53) Since k∈A⊆QNimplies ∥k∥1⩽N, Pr(k∈A)⩽ELh1(k∈A) \|QN\|qN N∞Y ℓ=1(1 +qℓ N)si =PrL(k∈A) \|QN\|qN N∞Y ℓ=1(1 +qℓ N)s.(54) Hence, we can bound Pr( k∈A) by PrL(k∈A) times a factor depending only onNands, which is further bounded by the following lemma: Lemma 11. When N⩾1, 1 \|QN\|qN N∞Y ℓ=1(1 +qℓ N)s⩽AsN1/4 with Asa constant depending	https://arxiv.org/abs/2504.19138v1
on s. Proof. First we write ∞Y ℓ=1(1 +qℓ N)s= exp s∞X ℓ=1log(1 + qℓ N) . 40 Because log(1 + qℓ N) is monotonically decreasing in ℓ, ∞X ℓ=1log(1 + qℓ N)⩽Z∞ 0log 1 + exp( −πℓp s/12N) dℓ =1 πr 12N sZ∞ 0log 1 + exp( −ℓ) dℓ =πr N 12s. Hence∞Y ℓ=1(1 +qℓ N)s⩽exp πr sN 12 . Our conclusion then follows from equation (15) and qN N= exp( −πp sN/12). Now we are ready to prove Lemma 7. Proof of Lemma 7. Equation (54) and Lemma 11 imply Pr \|κj\|p λN/s−2 > ϵ ⩽AsN1/4PrL \|κj\|p λN/s−2 > ϵ . (55) Because \|κj\|=X ℓ∈N1{ℓ∈κj}=X ℓ∈NXjℓ, we have EL[\|κj\|] =X ℓ∈Nqℓ N 1 +qℓ N=X ℓ∈Nexp(−πℓp s/12N) 1 + exp( −πℓp s/12N), VarL(\|κj\|) =X ℓ∈Nqℓ N (1 +qℓ N)2=X ℓ∈Nexp(−πℓp s/12N) (1 + exp( −πℓp s/12N))2. Since qℓ N/(1 +qℓ N) is monotonically decreasing in ℓ, Z∞ 1exp(−πℓp s/12N) 1 + exp( −πℓp s/12N)dℓ⩽EL[\|κj\|]⩽Z∞ 0exp(−πℓp s/12N) 1 + exp( −πℓp s/12N)dℓ. Recall that λ= 3 log(2)2/π2. The difference between the above two integral is O(1) and Z∞ 0exp(−πℓp s/12N) 1 + exp( −πℓp s/12N)dℓ=log(2) πr 12N s= 2r λN s, so EL[\|κj\|]∼2r λN s+O(1). (56) 41 Similarly, because qℓ N<1 and x/(1 +x2) is monotonically increasing in xover [0,1], we know qℓ N/(1 +qℓ N)2is monotonically decreasing in ℓoverℓ⩾0 and Z∞ 1exp(−πℓp s/12N) (1 + exp( −πℓp s/12N))2dℓ⩽VarL(\|κj\|)⩽Z∞ 0exp(−πℓp s/12N) (1 + exp( −πℓp s/12N))2dℓ. The difference is again O(1) and Z∞ 0exp(−πℓp s/12N) (1 + exp( −πℓp s/12N))2dℓ=1 πr 3N s, so VarL(\|κj\|)∼1 πr 3N s+O(1). (57) By Bernstein’s inequality, PrL \|κj\| −EL[\|κj\|] > t ⩽2 exp −t2/2 VarL(\|κj\|)+t/3 for any t >0. Setting t=ϵp λN/4s, we get PrL \|κj\| −EL[\|κj\|] > ϵq λN 4s ⩽2 exp −ϵ2λN/8s VarL(\|κj\|)+ϵ√ λN/4s/3 or equivalently PrL \|κj\|p λN/s−EL[\|κj\|]p λN/s >ϵ 2 ⩽2 exp −r λN sϵ2/8 VarL(\|κj\|)/p λN/s +ϵ/6 . Because ϵ <1 and VarL(\|κj\|)∼π−1p 3N/s, the right hand side can be bounded by 2 exp( −Bsϵ2√ N) for some Bs>0. If further \|EL[\|κj\|]/p λN/s−2\|< ϵ/2, PrL \|κj\|p λN/s−2 > ϵ ⩽PrL \|κj\|p λN/s−EL[\|κj\|]p λN/s >ϵ 2 ⩽2 exp(−Bsϵ2√ N) and we have proven equation (23) in view of equation (55). On the other hand, if\|EL[\|κj\|]/p λN/s−2\|⩾ϵ/2, equation (56) implies ϵ2√ N⩽4p λ/s EL[\|κj\|]−2r λN s =O(√s). So by decreasing Bsif necessary, we can assume Bsϵ2√ N⩽1 for any ϵsatisfying \|EL[\|κj\|]/p λN/s −2\|⩾ϵ/2. After increasing Asif necessary so that As⩾ exp(1), AsN1/4exp(−Bsϵ2√ N)⩾Asexp(−1)⩾1 and equation (23) is trivially true. 42 The proofs of Lemma 5 and Lemma 9 are similar. By an abuse of notation, we let Pr and PrLbe the probability when k1, ...krare sampled independently fromU(QN) and L(k), respectively. An analogous argument using the impor- tance sampling trick shows for any subset A⊆(QN)r Pr((k1, . . . ,kr)∈A)⩽PrL((k1, . . . ,kr)∈A)1 \|QN\|qN N∞Y ℓ=1(1 +qℓ N)sr .(58) Proof of Lemma 5. To simplify our notation, we write k⊕=⊕r i=1kiwith com- ponents ( k⊕ 1, . . . , k⊕ s). By equation (58) and Lemma 11, Pr k⊕∈QN ⩽Ar sNr/4PrL k⊕∈QN ⩽Ar sNr/4PrL ∥k⊕∥1⩽N .(59) By the definition of	https://arxiv.org/abs/2504.19138v1
k⊕,X⊕ jℓ=1{ℓ∈κ⊕ j}equals 1 if and only if ℓ∈κjfor an odd number of kamong k1, ...,kr. By a binomial distribution with success probability qℓ N/(1 +qℓ N), PrL(X⊕ jℓ= 1) =⌈r/2⌉X j=1r 2j−1qℓ N 1 +qℓ N2j−11 1 +qℓ Nr−2j+1 =1 2−1 21−qℓ N 1 +qℓ Nr . (60) Also notice that {X⊕ jℓ, j∈1:s, ℓ∈N}are jointly independent under L(k) and ∥k⊕∥1=sX j=1X ℓ∈NℓX⊕ jℓ. By Markov’s inequality, for any t >0 PrL ∥k⊕∥1⩽N = PrL exp −tsX j=1X ℓ∈NℓX⊕ jℓ ⩾e−tN ⩽etNELh exp −tsX j=1X ℓ∈NℓX⊕ jℓi =etNsY j=1Y ℓ∈N 1−PrL(X⊕ jℓ= 1)(1 −e−tℓ) ⩽exp tN−sX ℓ∈NPrL(X⊕ jℓ= 1)(1 −e−tℓ) . (61) 43 Because PrL(X⊕ jℓ= 1) is monotonically increasing in randr⩾2, X ℓ∈NPrL(X⊕ jℓ= 1)(1 −e−tℓ)⩾X ℓ∈N1 2−1 21−qℓ N 1 +qℓ N2 (1−e−tℓ) =X ℓ∈N1 2(1−e−tℓ)(1 +qℓ N)2−(1−qℓ N)2 (1 +qℓ N)2 =X ℓ∈N2(1−e−tℓ)qℓ N (1 +qℓ N)2. Setting t=−αlog(qN) for α >0 that we will tune later, we have X ℓ∈N2(1−e−tℓ)qℓ N (1 +qℓ N)2= 2X ℓ∈Nqℓ N (1 +qℓ N)2−2X ℓ∈Nqαℓ Nqℓ N (1 +qℓ N)2 Similar to equation (57), because both qℓ N/(1 +qℓ N)2andqαℓ Nqℓ N/(1 +qℓ N)2are monotonically decreasing in ℓoverℓ⩾0, X ℓ∈Nqℓ N (1 +qℓ N)2∼Z∞ 0exp(−πℓp s/12N) (1 + exp( −πℓp s/12N))2dℓ+O(1) =1 πr 12N sZ∞ 0exp(−ℓ) (1 + exp( −ℓ))2dℓ+O(1) and X ℓ∈Nqαℓ Nqℓ N (1 +qℓ N)2∼Z∞ 0exp(−π(α+ 1)ℓp s/12N) (1 + exp( −πℓp s/12N))2dℓ+O(1) =1 πr 12N sZ∞ 0exp(−(α+ 1)ℓ) (1 + exp( −ℓ))2dℓ+O(1). Combining t=−αlog(qN) =απp s/12Nwith the above equations, we get tN−sX ℓ∈NPrL(X⊕ jℓ= 1)(1 −e−tℓ)⩽tN−2sX ℓ∈Nqℓ N−qαℓ Nqℓ N (1 +qℓ N)2∼c(α)√ sN+O(s) ifc(α)̸= 0 with c(α) =απ√ 12−4√ 3 πZ∞ 0exp(−ℓ)−exp(−(α+ 1)ℓ) (1 + exp( −ℓ))2dℓ. Because c(α)→ ∞ asα→ ∞ and c′(α) =π√ 12−4√ 3 πZ∞ 0ℓexp(−(α+ 1)ℓ) (1 + exp( −ℓ))2dℓ 44 is strictly increasing in α, we see c(α) has a unique minimum α∗over α⩾0. Furthermore, c′(0) =π√ 12−4√ 3 πZ∞ 0ℓexp(−ℓ) (1 + exp( −ℓ))2dℓ=π√ 12−4√ 3 log(2) π<0, soα∗>0 and c(α∗)<0. A numerical approximation using Mathematica shows α∗≈0.24 and c(α∗)<−0.066. By choosing t=−α∗log(q), we have shown exp tN−sX ℓ∈NPrL(X⊕ jℓ= 1)(1 −e−tℓ) ⩽exp c(α∗)√ sN+O(s) . Putting together equation (59) and equation (61), we get Pr k⊕∈QN ⩽Ar sNr/4exp c(α∗)√ sN+O(s) . For a threshold Rs⩾2 that we will determine later, we can choose Bssmall enough so that Bslog(Rs)⩽−c(α∗)√s. By increasing Asif necessary to ac- count for the exp( O(s)) term, equation (18) holds for all N⩾1 and r⩽Rs. It remains to show equation (18) holds for some As, Bs>0 when r > R s for some threshold Rs. Let ℓ∗be the largest ℓ∈Nfor which PrL(X⊕ jℓ= 1)⩾ 1/4. We conventionally set ℓ∗= 0 if PrL(X⊕ jℓ= 1) <1/4 for all ℓ∈N. By equation (60), PrL(X⊕ jℓ= 1) =1 2−1 21−qℓ N 1 +qℓ Nr =1 2−1 21−exp(−πℓp s/12N) 1 + exp( −πℓp s/12N)r . Because PrL(X⊕ jℓ= 1) is monotonically decreasing in ℓoverℓ⩾0,ℓ∗equals the floor of the solution of PrL(X⊕ jℓ= 1) = 1 /4. A straightforward calculation gives ℓ∗=⌊log1 + 2−1/r 1−2−1/rr 12N π2s⌋. By convexity of the function f(x) =x−1/r, 2−1−1/rr−1⩽1−2−1/r⩽r−1. Hence r⩽1 + 2−1/r 1−2−1/r⩽(1 +	https://arxiv.org/abs/2504.19138v1
2−1/r)21+1/rr⩽8r and log(r)r 12N π2s−1⩽ℓ∗⩽log(8 r)r 12N π2s. (62) log(r)r 12N π2s−1⩽ℓ∗⩽log(8 r)r 12N π2s. 45 Using the inequality 1 −exp(−x)⩾x/(1+x) when x⩾0, equation (61) becomes PrL ∥k⊕∥1⩽N ⩽exp tN−sX ℓ∈NPrL(X⊕ jℓ= 1)tℓ 1 +tℓ ⩽exp tN−sX ℓ∈N1{ℓ⩽ℓ∗}tℓ 4(1 + tℓ) = exp tN−stℓ∗(ℓ∗+ 1) 8(1 + tℓ∗) . (63) Setting t=p s/N, we derive from equation (62) that there exists a large enough Rsso that for all r⩾Rs, 1 +tℓ∗⩽1 + log(8 r)r 12 π2<2 log( r)r 12 π2 and stℓ∗(ℓ∗+ 1)⩾√ sNlog(r)r 12 π2 log(r)r 12 π2−rs N >√ sNlog(r)26 π2. By increasing Rsif necessary, we further have for all r⩾Rs tN−stℓ∗(ℓ∗+ 1) 8(1 + tℓ∗)⩽√ sN−√ sNlog(r)√ 3 16π<−√ sNlog(r)√ 3 32π. Putting together equation (59) and equation (63), we get for r⩾Rs Pr k⊕∈QN ⩽Ar sNr/4exp(−√ sNlog(r)√ 3 32π), which completes the proof. Proof of Lemma 9. By equation (54) and Lemma 11, Pr κ>ρ√ N j,1 =∅ ⩽AsN1/4PrL κ>ρ√ N j,1 =∅ . Because κ>ρ√ N j,1 =∅if and only if ℓ /∈κj,1for all ℓ > ρ√ N, PrL κ>ρ√ N j,1 =∅ =∞Y ℓ=⌈ρ√ N⌉1 1 +qℓ N= exp −∞X ℓ=⌈ρ√ N⌉log(1 + qℓ N) . Because log(1 + qℓ N) is monotonically decreasing in ℓ, ∞X ℓ=⌈ρ√ N⌉log(1 + qℓ N)⩾Z∞ ⌈ρ√ N⌉log 1 + exp( −πℓp s/12N) dℓ ⩾cρ,s√ N−log(2) (64) 46 for cρ,s=1 πr 12 sZ∞ πρ√ s/12log 1 + exp( −ℓ) dℓ. Hence Pr κ>ρ√ N j.1 =∅ ⩽2AsN1/4exp −cρ,s√ N . Similarly, by equation (58) and Lemma 11, Pr κ>ρ√ N j,1 =κ>ρ√ N j,2 ⩽A2 sN1/2PrL κ>ρ√ N j,1 =κ>ρ√ N j,2 . Because κ>ρ√ N j,1 =κ>ρ√ N j,2 if and only if each ℓ > ρ√ Neither appears in both of or neither of κj,1, κj,2, PrL(κ>ρ√ N j,1 =κ>ρ√ N j,2) =∞Y ℓ=⌈ρ√ N⌉1 +q2ℓ N (1 +qℓ N)2 = exp∞X ℓ=⌈ρ√ N⌉log(1 + q2ℓ N)−∞X ℓ=⌈ρ√ N⌉2 log(1 + qℓ N) . Again by monotonicity of log(1 + q2ℓ N), ∞X ℓ=⌈ρ√ N⌉log(1 + q2ℓ N)⩽Z∞ ⌈ρ√ N⌉−1log 1 + exp( −2πℓp s/12N) dℓ ⩽c′ ρ,s√ N+ log(2) for c′ ρ,s=1 2πr 12 sZ∞ 2πρ√ s/12log 1 + exp( −ℓ) dℓ. Notice that c′ ρ,s< cρ,s/2. Along with equation (64), we get the bound Pr κ>ρ√ N j,1 =κ>ρ√ N j,2 ⩽8A2 sN1/2exp −2(cρ,s−1 2c′ ρ,s)√ N . Our conclusion follows by taking Aρ,s= 2√ 2AsandBρ,s=cρ,s−c′ ρ,s/2> (3/4)cρ,s. 47	https://arxiv.org/abs/2504.19138v1
Optimal experimental design for parameter estimation in the presence of observation noise Jie Qi1and Ruth E. Baker2 1College of Information Science and Technology, Donghua University, Shanghai, China 2Mathematical Institute, University of Oxford, Oxford, United Kingdom Abstract Using mathematical models to assist in the interpretation of experiments is becoming increasingly important in research across applied mathematics, and in particular in biology and ecology. In this context, accurate parameter estimation is crucial; model parameters are used to both quantify observed behaviour, characterise behaviours that cannot be directly measured and make quantitative predictions. The extent to which parameter estimates are constrained by the quality and quantity of available data is known as parameter identifiability, and it is widely understood that for many dynamical models the uncertainty in parameter estimates can vary over orders of magnitude as the time points at which data are collected are varied. Here, we use both local sensitivity measures derived from the Fisher Information Matrix and global measures derived from Sobol’ indices to explore how parameter uncertainty changes as the number of measurements, and their placement in time, are varied. We use these measures within an optimisation algorithm to determine the observation times that give rise to the lowest uncertainty in parameter estimates. Applying our framework to models in which the observation noise is both correlated and uncorrelated demonstrates that correlations in observation noise can significantly impact the optimal time points for observing a system, and highlights that proper consideration of observation noise should be a crucial part of the experimental design process. 1arXiv:2504.19233v1 [math.ST] 27 Apr 2025 1 Introduction Mathematical modelling serves as a crucial technique in the interpretation of experiments and offers conceptual insights into the mechanisms underlying complex systems across various fields, particularly in biology and ecology. In this context, accurate parameter estimation is essential; model parameters are now routinely employed to quantify observed behaviours, characterize behaviours that cannot be directly measured, and make quantitative predictions. The extent to which parameter estimates are constrained by the quality and quantity of available data is termed parameter identifiability. For many dynamical models, the uncertainty in parameter estimates can vary over orders of magnitude as the time points at which data are collected are varied. Consequently, optimising data collection strategies to improve parameter identifiability is a crucial aspect of experimental design, particularly for biological systems that frequently exhibit complex and non-linear behaviours. Optimal experimental design methodologies provide a means to optimise experimental pro- tocols to e.g., maximise information acquisition or minimise uncertainty in parameter estimates under resource constraints. The choice of objective function in optimal experimental design is crucial as these set specific optimality criteria, and sensitivity measures are commonly employed. In particular, local sensitivity measures incorporated in the Fisher information matrix [1, 2, 3, 4] are often used because the inverse of the Fisher information matrix provides an estimate of the lower bound of the covariance matrix of the estimated parameters, based on the Cram´ er-Rao inequality [5]. Local sensitivity measures assume a linear relationship between parameter pertur- bations and model responses locally in the region of the specified parameter	https://arxiv.org/abs/2504.19233v1
values. A drawback is therefore that local sensitivity measures can lead to inefficient experimental designs if the input parameters are not close to their “true” values. To overcome the limitations of local sensitivity measures, global sensitivity analyses have been increasingly adopted in optimal ex- perimental design [6]. These measures consider nonlinear effects, non-monotonic behaviour and dependencies between multiple parameters [7], making them more suited to the complexities of biological systems. In addition, global sensitivities take parameter ranges as inputs, enhancing the robustness of the experimental design process [8, 9]. A key unexplored question in the literature is the extent to which changes in the character- istics of the measurement / observation noise process impact the output of optimal experimen- tal design protocols. As such, the goal of this paper is to apply optimal experimental design methodologies to a widely used model in mathematical biology to determine optimal observation schemes, specifically focussing on the time points at which data are collected and how different types of noise influence optimal data collection strategies. We will explore the ability of both lo- 2 cal and global measures, used within experimental design approaches, to reduce the uncertainty in parameter estimates in the face of observation noise. The model we consider is an ordinary differential equation model, as is commonly used to describe dynamical systems in biology, and the observations are assumed to consist of the solution of this model plus an “observation noise”. In this work, our assumption is that that the observation noise accounts for factors not explicitly included in the model, for example, stochasticity in the underlying processes or measurement errors. Most modelling studies in mathematical biology assume that the observation noise is independent Gaussian distributed. This assumption simplifies much statistical analysis, but it does not hold universally across all applications [10, 11]. Although there has been some research into the impact of autocorrelated observation noise processes on parameter estimation [10, 12, 13], the extent to which it affects optimal experimental design remains under-explored. As such, the overarching aim of this work is to understand the extent to which changes in the observation noise process impact our ability to estimate model parameters and the “optimal” experimental set-up that minimises the uncertainty in parameter estimates. In particular, in many applications there may be correlations between adjacent observation error terms in time series models; these may be caused by, for example, measurement equipment biases or model misspecification [10, 11]. Autocorrelated noise can be represented both through discrete-time models, such as autoregressive, moving average, autoregressive moving average and autoregressive integrated moving average processes [14], and continuous-time processes, such as fractional Brownian motion [15], Cox-Ingersoll-Ross processes [16] and Ornstein-Uhlenbeck (OU) [17] processes. The OU process, a continuous-time analogue of the discrete-time autore- gressive process of order 1, is particularly suited for analysing the impact of autocorrelated noise on parameter estimation because the extent of the autocorrelation between time points depends solely on the magnitude of the time between them. Therefore, in this study, we use the OU process to model autocorrelated observation noise. This paper explores optimal experimental	https://arxiv.org/abs/2504.19233v1
design within the context of both uncorrelated (in- dependent, identically distributed (IID)) and autocorrelated (OU distributed) observation noise, specifically focussing on the logistic model, which is ubiquitously applied in the modelling of biological systems. We suggest an approach to optimal experimental design to minimise param- eter uncertainty, utilising both local sensitivity measures derived from the Fisher information matrix and global sensitivity measures derived from Sobol’ indices as objective functions, and we use our approach to show that changes in the structure of the observation error model can significantly impact the optimal experimental design. 3 The paper is structured as follows. Section 2 introduces the relevant methodology, including the logistic model and the information measures and noise processes to be used in this work. It also describes formulation of the optimal experimental design problem and an overview of the profile-likelihood approach for estimating parameters and confidence intervals. The main results, including the impact of the specific noise process on parameter estimates and the optimal observation schemes, are given in Section 3. The paper concludes with a brief discussion in Section 4. 2 Methods In this section, we outline the mathematical model and relevant statistical techniques used throughout the paper. All relevant code to implement the techniques and reproduce the figures in this work is provided on Github at https://github.com/Jane0917728/Optimal-Experim ental-Design-and-Parameter-Estimation-with-Autocorrelated-Observation-Noise . 2.1 Mathematical model We consider the logistic model for population growth, dC(t) dt=rC(t)/parenleftbigg 1−C(t) K/parenrightbigg , (1) where C(t) is the population at time t,r >0 is the growth rate, K > 0 is the carrying capacity, with K= lim t→∞C(t), and C(0) = C0>is the initial population size. For C0≪K, population growth is characterised by an initial exponential growth phase in which resources are assumed abundant, a deceleration phase where competition for resources becomes important, and a steady state phase in which the population stabilises at the carrying capacity. The logistic model can be solved explicitly to give C(t) =C0K (K−C0)e−rt+C0. (2) In the model, three parameters θ= (θ1, θ2, θ3) = ( r, K, C 0) are to be estimated, and we will often write C(t;θ) to highlight the dependence of the model output on the parameters. The “true” parameters are denoted by θ∗= (r∗, K∗, C∗ 0). Throughout this work, we set r= 0.2, K= 50 and C0= 4.5 and we consider the model up to a final time tfinal= 80, which is sufficient for the population to be very close to carrying capacity. 4 We assume that it is possible to observe the state of the system at ns(strictly positive) discrete observation times ,t1, . . . , t ns, and that observations are of the form Y=C(θ∗) +ϵ, (3) where Y= [Y(t1), . . . , Y (tns)], (4) C(θ∗) = [C(t1;θ∗), . . . , C (tns;θ∗)], (5) ϵ= [ϵ(t1), . . . , ϵ (tns)], (6) represent the vector forms of the observations, models outputs and measurement noise at discrete observation times t1, . . . , t ns, respectively. 2.2 Measures of information To quantify the amount of information about the parameters contained in	https://arxiv.org/abs/2504.19233v1
a dataset, we will consider both the Fisher information matrix, which provides a local measure of uncertainty, and Sobol’ indices, which provide a global measure of uncertainty. Fisher information matrix. The Fisher information matrix can be defined using the expec- tation of the Hessian of the log-likelihood function as [1, 2] F={Fij}=/braceleftbigg −E/bracketleftbigg∂2L(θ\|Y) ∂log(θi)∂log(θj)/bracketrightbigg/bracerightbigg , (7) where E[·] refers to the expectation operator and L(θ\|Y) is the log-likelihood of parameters θgiven data Y. The inverse of the Fisher information matrix offers a lower bound on the covariance matrix for unbiased estimators via the Cram´ er-Rao bound [5]. As such, maximising an objective function based on the Fisher information matrix is equivalent to minimising the uncertainty of the estimated parameters governed by the covariance matrix. Sobol’ indices. Sobol’ indices are based on the idea of variance decomposition and evaluate the contribution of each parameter’s variance to the total variance of the measurements [6, 18, 19]. They are particularly suitable for measuring the sensitivity of complex models where multiple parameters influence the measurements, and the relationships between them are non- linear or involve interactions among parameters. 5 Given a model of the form M=M(t;θ), where θ= (θ1, . . . , θ p) (recall that here that p= 3 with θ1=r,θ2=Kandθ3=C0), the total-effect Sobol’ index is defined, at time t, as Si(t) =Eθ∼i[Vθi[M(t;θ)\|θ∼i]] V[M(t;θ)],fori= 1, . . . , p, (8) where Vis the total variance, Vθidenotes the variance over all possible values of θi, and the inner expectation operator Eθ∼i[M(t;θ)\|θi] indicates, given θifixed, the mean of M(t;θ) over all possible values of θ∼i, the set of the parameters excluding θi.Si(t) measures the total effect ofθion the model output M(t;θ), including both its direct influence and its interactions with other parameters, and it is normalised such that Si(t)∈[0,1]. A high value of Si(t) indicates that the ithparameter has a strong effect on the output. 2.3 Independent and identically distributed observation noise For IID Gaussian observation noise we have ϵ∼ N (0, σ2 IIDI), where σ2 IID>0 is the noise variance, so that Y∼ N(C(θ∗), σ2 IIDI) and the log-likelihood function is given by L(θ\|Y) =−ns 2log(2πσ2 IID)−1 2σ2 IID(Y−C(θ))(Y−C(θ))T. (9) Applying maximum likelihood estimation to estimate the noise variance gives ˆσ2 IID=1 ns(Y−C(θ))(Y−C(θ))T=1 nsns/summationdisplay s=1(Y(ts)−C(ts;θ))2. (10) Throughout this work we will fix the estimate for σ2 IIDto be the maximum likelihood estimate (MLE) ˆ σ2 IIDgiven in Equation (10), and focus on optimising the measurement times to best estimate the model parameters θ. To calculate the Fisher information matrix, F, we first calculate ∂2L(θ\|Y) ∂log(θi)∂log(θj)=−1 σ2 IIDθiθj∂C ∂θi/parenleftbigg∂C ∂θj/parenrightbiggT +1 σ2 IIDθiθj(Y−C(θ))/parenleftbigg∂2C ∂θi∂θj/parenrightbiggT . (11) Exploiting the fact that E[Y−C(θ)] = 0, the elements of the Fisher information matrix can be written Fij=1 ˆσ2 IIDθiθj∂C ∂θi/parenleftbigg∂C ∂θj/parenrightbiggT . (12) 6 We construct a global information matrix, G, for IID Gaussian observation noise based on the total effect Sobol’ indices with entries Gij=SiST j=ns/summationdisplay s=1Si(ts)Sj(ts), (13) where Si= [Si(t1), . . . , S i(tns)] is a row vector composed of the total effect Sobol’ indices of parameter iat each observation time ts,s= 1, .	https://arxiv.org/abs/2504.19233v1
. . , n s, as defined in Equation (8). Sobol’ indices are usually computed using Monte Carlo simulation [8, 19]. In this paper, we compute the Sobol’ indices directly using the Global Sensitivity Analysis Toolbox for Matlab [20]. 2.4 Autocorrelated observation noise We use an OU process, as defined by the stochastic differential equation dϵ(t) =−ϕ ϵ(t)dt+σOUdW(t), ϵ(0) = ϵ0, (14) to model autocorrelated measurement noise, with ϵ0∼ N(0, σ2 OU/(2ϕ)). In Equation (14), ϕ >0 is the mean-reversion rate and σOU>0, often known as the volatility coefficient, controls the intensity of the random fluctuations introduced by the Wiener process W(t). The solution to Equation (14) is given by ϵ(t) =ϵ0e−ϕt+σOU/integraldisplayt 0e−ϕ(t−τ)dW(τ), (15) where the second term follows N(0, σ2(1−e−2ϕt)/(2ϕ)). Hence, ϵ(t) is conditionally normally distributed with E[ϵ(t)] =E[ϵ0]e−ϕt= 0, (16) V[ϵ(t)] =e−2ϕtV[ϵ0] +V/bracketleftbigg σOU/integraldisplayt 0e−ϕ(t−τ)dW(τ)/bracketrightbigg =σ2 OU 2ϕ. (17) Note that the choice of ϵ(0)∼ N(0, σ2 OU/(2ϕ)) means that ϵ(t) follows the stationary distribution at all times. We use the OU process described above to model autocorrelated observation noise with data collected at observation time points t1, . . . , t ns. We set ϵ= [ϵ(t1), ϵ(t2), . . . , ϵ (tns)] and use the fact that ϵ(t1)∼ N(0, σ2 OU/(2ϕ)), with Equation (14) describing the noise at subsequent times t2, . . . , t ns. Given Equation (15), it follows that for i= 1, . . . , n s−1 the conditional distribution 7 ofϵ(ti+1) given ϵ(ti) is ϵ(ti+1)\|ϵ(ti)∼ N/parenleftbigg ϵ(ti)e−ϕ∆i,σ2 OU 2ϕ/parenleftig 1−e−2ϕ∆i/parenrightig/parenrightbigg , (18) where ∆ i=ti+1−tiis the time between consecutive observations. The autocorrelation is given by AC(∆ i) =e−ϕ∆ifor time lag ∆ i. Hence we see that the autocorrelation diminishes as the time interval between observations increases, and that the smaller the value of ϕ, the stronger the autocorrelation. The likelihood function can be expressed as L(ϵ) =−ns 2log/parenleftbigg πσ2 OU ϕ/parenrightbigg −ns−1/summationdisplay i=11 2log/parenleftig 1−e−2ϕ∆i/parenrightig −ϕ σ2 OUϵ2(t1)−ϕ σ2 OUns−1/summationdisplay i=1/parenleftbig ϵ(ti+1)−ϵ(ti)e−ϕ∆i/parenrightbig2 1−e−2ϕ∆i, (19) or, equivalently, L(ϵ) =−ns 2log(2π)−1 2log(det( Σ))−1 2ϵΣ−1ϵT, (20) where Σ∈Rns×nsis the covariance matrix of the OU process, with entries Σij= Cov[ ϵ(ti), ϵ(tj)] =σ2 OU 2ϕe−ϕ\|ti−tj\|, (21) fori, j= 1, . . . , n s. Given that ϵ=Y−C(θ), we can write L(θ\|Y) =−ns 2log(2π)−1 2log(det( Σ))−1 2(Y−C(θ))Σ−1(Y−C(θ))T. (22) To avoid issues with practical identifiability, we assume that ϕis known, and we apply maximum likelihood estimation to estimate the volatility coefficient as ˆσ2 OU=2ϕ ns(Y−C(θ))˜Σ−1(Y−C(θ))T, (23) where the entries of ˜Σare ˜Σij=e−ϕ\|ti−tj\|. (24) We note that σ2 OUdoes not affect the process of optimising the experimental design as it serves merely as a scaling factor in the objective function. 8 The entries of the Fisher information matrix, F, can be written Fi,j=θiθj∂C ∂θiΣ−1/parenleftbigg∂C ∂θj/parenrightbiggT , (25) and entries of the global sensitivity matrix, G, are defined as Gij=SiΣ−1ST j, (26) where Σis as defined in Equation (21) and, as before, Si= [Si(t1), . . . , S i(tns)] is a row vector composed of the total effect Sobol’ indices of parameter iat each time ts,s= 1, . . . , n s, as defined in Equation	https://arxiv.org/abs/2504.19233v1
(8). It is worth noting that the inverse of Σ, also known as the precision matrix, contains information about the conditional dependencies between different time points, and so their inclusion in the sensitivity matrices FandGenable quantification of how parameter sensitivities are influenced by temporal correlations in the data. 2.5 Optimal experimental design The objective of optimal experimental design in this work is to improve the reliability of param- eter estimation through optimising the experimental conditions; in this work, we will focus on optimising the observation times t1, . . . , t ns. Fisher information matrix. In the literature, there are three main quantities calculated from the Fisher information matrix that are routinely used to measure the quality of parameter estimates, namely the trace, the determinant and the minimum expected information gain [21, 22]. We found these metrics to yield similar results when used for optimizing the selection of observation times. Therefore, we choose to present results generated using the determinant of the Fisher information matrix. We denote the set of measurement times as t= [t1, . . . , t ns] so that the aim is to optimise max tf= max t/bracketleftig det(F(θ,t))/bracketrightig , (27) where F(θ,t) is the Fisher information matrix evaluated for parameters θat times t. We impose the following constraints on t: t1≥0, t ns≤tfinal, t s+1−ts≥∆tfors= 1, . . . , n s−1, (28) where ∆tis the minimum time interval between observations that is practical for experimental 9 design (throughout this work we take ∆t= 2.0). This optimal experimental design problem is a non-convex, continuous nonlinear programming problem [23]. We solve it using the fmincon function in Matlab with the interior-point method. To mitigate the risk of the algorithm con- verging to local optima, we implement a random restart technique, which involves generating 50 different random initial guesses to serve as starting points for fmincon. Global information matrix. Since Sobol’ indices need to be calculated using a Monte Carlo method, it is prohibitively time-consuming to consider the same optimisation problem as that for the Fisher information matrix. To make progress, we instead establish a mixed-integer nonlinear programming problem. We first, as for the Fisher information matrix, define ∆tto be the minimum time interval between observations that is practical for experimental design (again, taking ∆t= 2.0 throughout this work). We then define a fine grid of kpotential observation times [ tp 1, . . . , tp k], with tp i=t+(i−1)∆tfori= 1, . . . , k −1, and tp k=tfinal, where kis the largest integer such that tp k−1≤tfinal. The optimal experimental design aim then is to select the ns< k time points that give rise to the optimal objective function, which is now det( G), where Gis the global information matrix defined in Equation (13) (uncorrelated noise) or Equation (26) (correlated noise). We formulate the optimisation problem using the decision variables I= [I1, . . . , I k], where Ii∈ {0,1}fori= 1, . . . , k indicates whether an observation is made at time point tior not. The aim is then to optimise max	https://arxiv.org/abs/2504.19233v1
Ig= max I/bracketleftig det(G(θ,I))/bracketrightig , (29) subject to the constraint k/summationdisplay i=1Ii=ns. (30) We solve the optimisation problem in Equations (29)–(30) using PlatEMO, an evolutionary multi-objective optimisation platform for Matlab [24, 25] and, in particular, we use the com- petitive swarm optimizer. Throughout this work we use the parameter ranges r∈[0.14,0.26], K∈[35,65] and C0∈[3.15,5.85]. 2.6 Profile likelihood approach In order to assess the performance of the observation time points selected by solving the optimal experimental design problem, we calculate the profile likelihood. The practical identifiability of a parameter can be determined by examining the shape and width of the profile likelihood function [26]. A parameter is considered practically identifiable if its profile likelihood is well- 10 defined, meaning it is unimodal with a single peak, and a finite confidence interval. Conversely, a flat or broad profile likelihood suggests that a wide range of parameter values yield similar fits to the data, thereby indicating that the parameter is not practically identifiable. To calculate the univariate profile likelihoods, we partition the parameter vector θinto the interest parameter, θi, and nuisance parameters, θ∼i. We define the univariate, normalised profile log-likelihood function for the interest parameter θi, lp(θi\|Y) = sup θ∼iL(θ\|Y)−sup θL(θ\|Y). (31) Given this definition, we can use lp(θi\|Y) = 1 .92 to define an approximate 95% confidence interval for θi[27], as follows CI =/braceleftbig θi\|lp(θi\|Y)≥ −χ2 0.95;1/2 =−1.92/bracerightbig , (32) where χ2 0.95;1represents the critical value of the chi-square distribution with one degree of freedom at the 95% confidence level. We use the fmincon function in Matlab to calculate the MLE, the profile likelihood and 95% confidence interval for each parameter. We use the 95% confidence intervals to generate prediction intervals for the model [28]. The prediction intervals are calculated through parameter sampling and log-likelihood evaluation: parameters are uniformly sampled and, for each sample, the normalised log-likelihood is com- puted and compared to a threshold of χ2 0.95;3/2. Parameters that exceed this threshold are retained, ensuring they lie within the 95% confidence region. From the valid samples, Mpop- ulation trajectories are generated at time points t= 1, . . . , n s, and the upper and lower bounds of these trajectories are calculated. Noise model bounds, based on the 5% and 95% quantiles of the relevant distribution, are then added into the upper and lower bounds, respectively, so as to form the prediction interval. 3 Results In this section we present the optimal experimental design results and the impact of optimally choosing measurement times on the parameter estimates. 3.1 Parameter identifiability under the different noise models We first explore the extent to which the model parameters can be estimated under different noise models when the data are sampled at regular time intervals, with 11 observations at times 11 Figure 1: Parameter identifiability under the different noise models, with data collected at 11 regularly spaced time points t= 0,8,16, . . . , 80. The left column shows the sampled data, the underlying model solution using the input parameters, and the solution generated using the MLE parameters. The shaded region shows the 95% prediction	https://arxiv.org/abs/2504.19233v1
interval generated using the 95% confidence intervals from the profile likelihoods. The right-most three panels show the profile likelihoods for each parameter. The results in the top row are generated using uncorrelated Gaussian noise with σ2 IID= 9.0. The results in the middle row are generated using correlated OU noise with ϕ= 0.02 and σ2 OU/(2ϕ) = 9 .0. The results in the bottom row are generated using a misspecified noise model: the data are generated using correlated OU noise with ϕ= 0.02 and σ2 OU/(2ϕ) = 9 .0, whilst the analysis is carried out assuming IID Gaussian noise with σ2 IID= 9.0. t= 0,8,16, . . . , 80. Results for ϕ= 0.02 are shown in Figure 1, with similar results for ϕ= 0.1 (less correlated noise) shown in Supplementary Figure S1. The left column shows the sampled data (circles) along with the underlying model solution using the input parameters (dashed line) and the solution generated using the MLE parameters (solid line), and the shaded region shows the 95% prediction interval. The top row shows the results from the uncorrelated noise model, the second row shows the results from the correlated noise model, whilst the bottom row shows the results in the case of misspecification in the noise model, where correlated noise was used to generate the observations but the profile likelihoods were generated using the uncorrelated noise 12 Figure 2: Change in the mean 95% confidence interval for parameters r,KandC0as the variance increases under both correlated and uncorrelated noise, and when the noise is misspecified (as per Figure 1). Top row: σ2=σ2 IID=σ2 OU/(2ϕ) with ϕ= 0.02 so that the variance is equivalent across the different noise models. Bottom row: σ2=σ2 IID=σ2 OUwith ϕ= 0.02 so that the variance is larger in the correlated noise process than the uncorrelated noise process. In all cases, results were generated by averaging the confidence interval width from 1000 simulations for each noise model and parameter set, and 11 observations were used. model. To enable a comparison of the effects of uncorrelated observation noise and correlated observation noise on parameter inference under consistent variance, we choose parameters such thatσ2 IID=σ2 OU/(2ϕ). We see that in all cases all model parameters are practically identifiable, with the widths of the confidence intervals depending on the both the noise model and the data. A key point to note is that incorrectly assuming uncorrelated noise can impact the parameters in different ways. For example, the carrying capacity, K, is predicted to be more confidently estimated, whilst the uncertainty in the growth rate is predicted much higher in the misspecified case. Figure 2 shows how the width of the confidence intervals changes with the variance for both uncorrelated and correlated noise, as well as misspecified noise1, in each case using 11 regularly spaced observation time points, t= 0,8,16, . . . , 80. The top row shows results with σ2=σ2 IID=σ2 OU/(2ϕ), and the mean reversion rate of the OU process held constant at ϕ= 0.02 1Recall that in the misspecified noise case, the data are generated using correlated	https://arxiv.org/abs/2504.19233v1
data but the confidence intervals are generated assuming the data are uncorrelated. 13 so that the noise variance is equivalent across both noise models. We see that, for all values of σ2, the confidence intervals for all parameters are almost identical for uncorrelated noise and misspecified noise, with greater confidence in the parameter estimated for the carrying capacity K, and less confidence in the estimates for the growth rate, r, and the initial condition, C0, compared to the correlated noise case. This plots highlights the potential pitfalls of incorrect assumptions on the noise model, in that it can potentially lead a modeller to either have too little, or too much, confidence in parameter estimates. The bottom row of Figure 2 illustrates that setting σ2 IID=σ2 OUwith significant noise correlation ( ϕ= 0.02) results in greatly reduced confidence in the parameter estimates for the correlated noise case compared to the uncorrelated and misspecified cases. This is to be expected as, in this case, the variance of the correlated noise process is 25 times larger than that of the uncorrelated noise process. 3.2 Optimal experimental design In this section, we use the framework outlined in Section 2.5 to explore how the optimal time points for observations change as the total number of observations, ns, is varied, and also how correlations in the observation noise influence the optimal measurement protocol. We use both the local measures obtained using the Fisher information matrix, and global measures obtained using the total Sobol’ index, as outlined in Section 2.2. All results are generated using ϕ= 0.02, which entails a significant degree of correlation in the noise process. Figure 3 shows the results of optimising the experimental design for uncorrelated noise, detailing the placement of the observations as the total number of observations is varied from three to ten. It also compares the confidence in parameter estimates under both optimised and evenly spread observations when five observations are made in total. We see that both the Fisher information matrix and the global information matrix lead to experimental designs that place the observations into three groups, with the first two groups of observations lying in the first 20 time units, which corresponds to the exponential growth phase, and the third group gathered towards the final time, which corresponds to the process being at steady state (carrying capacity). These results make sense intuitively, with the first two groups of time points enabling accurate inference of the growth rate, r, and the initial condition, C0, and the third group enabling accurate inference of the carrying capacity, K. These intuitive explanations are further supported by the plots in Figure 4 which shows how the sensitivity of the Fisher information matrix and global information matrix measures to variation in each parameter change over the course of the experiment. Our results also highlight how careful placement of the observations can enable accurate inferences to be made with fewer time points: the right-hand three plots 14 Figure 3: The results of optimising experimental design in the uncorrelated noise case as the number of observations, ns, is	https://arxiv.org/abs/2504.19233v1
varied from three to ten. The top row shows the results from evenly distributed observations, whilst the second row shows the optimal experimental design derived using the Fisher information matrix, and the bottom row shows the optimal experimental design derived using the global information matrix. In each case, the left-hand plots show the observations time points, whilst the right-hand three plots show the profile likelihoods obtained from five observations. In all plots, σ2 IID= 9.0. highlight that under the optimised protocols, all parameters can be accurately estimated using just five observations, whereas neither the growth rate, r, or the initial condition, C0, can be accurately inferred using five equally spaced observations. When the observation noise is correlated (see Figure 5), the optimisation results change dramatically: the majority of the observation points are evenly distributed during the first half of the experiment, until approximately 40 time units, with only a single observation in the second half of the experiment, which is generally placed at, or very close to, the terminal time, tfinal. This change in experimental design is intuitive: the significant correlation in the noise process (small ϕ) means that less information about the parameters is gained from two closely placed observations. Once again, we see that the Fisher information matrix and the global 15 Figure 4: (a) Fisher information sensitivity and (b) global information sensitivity, formulated in terms of the gradient of the output, C(t;θ), with respect to the individual parameters. We evaluate the sensitivity at tk= 0,2,4, . . . , 80, with the Fisher information sensitivity calcu- lated as θi∂C(tk)/∂θiand the global information sensitivity calculated as Si(tk), as defined in Equation (8). information matrix provide very similar results in terms of optimal placement of the observation time points. We also note that with correlated observation noise, all parameters can in fact be inferred from five observations without any optimisation—evenly spacing the time points provides closed confidence intervals for all model parameters—although optimisation clearly reduces the uncertainty in the parameter estimates. 3.3 Impact of optimal experimental design upon parameter identifiability In this section, we explore how changes in the number of observation time points impact the confidence in parameter estimates. The mean parameter confidence intervals, calculated using the profile likelihood method presented in Section 2.6, are shown in Figure 6. The top row displays results for the uncorrelated noise model, the second row for the correlated noise model, and the bottom row for the case of model misspecification, where correlated noise was used to generate observations, but profile likelihoods were computed assuming an uncorrelated noise model. Additional results, for different values of ϕ, are shown in Supplementary Information Figure S2. Figure 6 shows the intuitive result that, in all cases, parameter estimates become more confident as the number of observation time points is increased, and that optimising the timing of observations generally improves the confidence in parameter estimates compared to the case of evenly distributing the observation time points. The exception to this rule is the carrying capacity parameter, K, where the confidence intervals are often marginally smaller for evenly spaced	https://arxiv.org/abs/2504.19233v1
time points. This is a direct result of the fact that the optimisation algorithms place the majority of the observations at early times in order to accurately estimate the growth rate, r, 16 Figure 5: The results of optimising experimental design in the correlated noise case as the number of observations, ns, is varied from three to ten. The top row shows the results from evenly distributed observations, whilst the second row shows the optimal experimental design derived using the Fisher information matrix, and the bottom row shows the optimal experimental design derived using the global information matrix. In each case, the left-hand plots show the observations time points, whilst the right-hand three plots show the profile likelihoods obtained from five observations. In all plots, σ2 OU/(2ϕ) = 9 .0. and the initial condition, C0. Note that for small numbers of observation time points ( ns= 3,4) the confidence intervals are half open when the observation points are evenly distributed, hence we do not display a result in those cases. In addition, note that when the number of evenly distributed points increases from five to seven in Figure 6, the width of the confidence interval for the parameter rincreases. One possible explanation is that among the five evenly distributed points, some provide more information for estimating the parameter r. Close investigation reveals that this is indeed the case: observations placed close to t= 20 are important for the estimation of r(see Supplementary Figure S3). It is worth noting that for the uncorrelated noise model, the mean confidence intervals for the carrying capacity, K, generated using local and global approaches to optimising the observation time points overlap. This is because the selected 17 Figure 6: Mean parameter confidence intervals in the case of uncorrelated noise (top row), correlated noise (middle row) and misspecified noise (bottom row). In each case, we plot the mean confidence interval as a function of the number of observation points, ns. The plots in each row are generated used 1000 simulations with σIID= 0.8 (top row), σOU= 0.16 and ϕ= 0.02 (middle and bottom rows) so that σ2 IID=σ2 OU/(2ϕ) = 0 .64. observation points lie in the saturation region and are nearly identical across the methods (as shown in the left column of Figure 3). 3.4 Impact of the autocorrelation level on experimental design In this section, we will investigate how the mean-reversion rate ϕ, reflecting the level of auto- correlation in the observation noise, influences the optimised distribution of observation times. Figure 7 shows how the optimised observation times (generated using the Fisher information matrix) change as ϕincreased (which corresponds to decreasing the autocorrelation). For very 18 Figure 7: The optimal observation time points as a function of the mean-reversion rate, ϕ. Note that the degree of autocorrelation decreases as ϕincreases. high autocorrelation, the majority of measurement points are placed within the first half of the experiment in order that the growth rate, r, and initial condition, C0, can be accurately measured. As the degree of autocorrelation diminishes ( ϕis increased), a measurement point is placed at,	https://arxiv.org/abs/2504.19233v1
or close to, the terminal time, which allows estimation of the carrying capacity, K. With more total observations, ns, further points are added at the end of time interval for larger values of ϕ, which enables increases in the accuracy of estimation of the carrying capacity, K. Figure 8 demonstrates how the parameter confidence intervals vary as a function of the degree of autocorrelation. In the top row the variance, σ2 OU/(2ϕ), is held constant as ϕis varied and we see that there is a complicated relationship between the confidence interval width and ϕ. For very low values of ϕ, which correspond to highly correlated observation noise, the majority of the observation points are placed early in the experiment which leads to accurate estimates for the growth rate, r, and carrying capacity, C0, and far less confidence in estimates of the carrying capacity, K. As the autocorrelation decreases (with increasing ϕ), increased numbers of observations are placed in the second half of the experiment and confidence in estimates of the carrying capacity, K, increase significantly. The bottom row demonstrates the intuitive result that, for σOU= 0.3 held constant and increasing ϕ(which corresponds to decreasing the variance, σ2 OU/(2ϕ), of the noise process), the mean confidence interval widths for all parameters decreases. 19 Figure 8: The mean confidence interval width for parameters estimated using 11 optimised observation time points as the mean-reversion rate, ϕ, increases from 0 .01 to 2 .0. In the top row the variance is held constant at σ2 OU/(2ϕ) = 4 .0 whilst in the bottow row the variance changes with σOU= 0.3 held constant. In all cases, results were generated using 1000 simulations. 4 Discussion This paper investigates the influence of different types of observation noise on optimal exper- imental design, where the aim is to determine the optimal observation times for accurate and confident parameter inference. Specifically, we analyze both uncorrelated (IID, Gaussian) and correlated (OU) noise and apply local techniques based on the Fisher information matrix, as well as global sensitivity approaches based on Sobol’ indices, to optimise observation times for the the logistic growth model. Our results highlight several key points. Firstly, the uncertainty in parameter estimates depends on the type of observation noise, with some model parameters becoming more identifi- able, and others less so, as the noise model is varied. Second, we demonstrate that, under both uncorrelated and correlated noise, optimising the placement of observation time points reduces the uncertainty in parameter estimates compared to na¨ ıve placement of measurement points and that, in many cases, this allows the practitioner to confidently estimate parameter values using fewer observations. Thirdly, our results show that the optimal observation time points are sensitive to the degree of noise correlation, with high autocorrelation favouring the majority of 20 time points in the first half of the experiment. On the whole, are results are relatively insensitive to the choice of objective function, though we highlight that use of the Fisher information matrix requires selection of parameter values a priori where as the global information matrix takes parameter ranges	https://arxiv.org/abs/2504.19233v1
as inputs, which means that it may be more appropriate for cases in which parameter values are relatively uncertain. The trade-off is the increased computational complexity of the global method compared to that of the local method. Our methodology is very general, in the sense that it can be applied in any context where it is possible to explicitly write down the likelihood and evaluate the associated Fisher and global information matrices. For example, it can be easily applied in the context of ordinary and partial differential equation-based models, and with a huge range of observation noise models. It would also be possible to integrate a cost-benefit analysis into the framework, for example using multi-objective optimisation to balance the costs of each observation against the quality of parameter estimates. In the future, we aim to extend our methodology to other models in mathematical biology to uncover more general insights regarding the effects of autocorrelated measurement processes on parameter estimates for differential equation-based models. Developing a method to diag- nose whether noise is correlated or independent would be beneficial. Additionally, it would be promising to optimize other experimental conditions, such as external inputs, beyond just measurement points, or to consider multi-objective optimal experimental design problems. Acknowledgements J.Q. would like to thank the Mathematical Institute, University of Oxford, for their support and hospitality during a visit to Oxford. R.E.B. acknowledges support from the Simons Foundation (MP-SIP-00001828). References [1] Gutenkunst R, Waterfall J, Casey F, Brown K, Myers C, Sethna J. 2007 Universally sloppy parameter sensitivities in systems biology. PLOS Computational Biology 3, 1871–1878. [2] Chis OT, Villaverde AF, Banga JR, Balsa-Canto E. 2016 On the relationship between sloppiness and identifiability. Mathematical Biosciences 282, 147–161. 21 [3] Telen D, Vercammen D, Logist F, Van Impe J. 2014 Robustifying optimal experiment design for nonlinear, dynamic (bio) chemical systems. Computers & Chemical Engineering 71, 415–425. [4] Zhang JF, Papanikolaou NE, Kypraios T, Drovandi CC. 2018 Optimal experimental design for predator–prey functional response experiments. Journal of The Royal Society Interface 15, 20180186. [5] Walter E, Pronzato L, Norton J. 1997 Identification of Parametric Models From Experi- mental Data vol. 1. Springer. [6] Schenkendorf R, Xie X, Rehbein M, Scholl S, Krewer U. 2018 The impact of global sen- sitivities and design measures in model-based optimal experimental design. Processes 6, 27. [7] Iooss B, Lemaˆ ıtre P. 2015 A review on global sensitivity analysis methods. In Dellino G, Meloni C, editors, Uncertainty Management in Simulation-Optimization of Complex Systems: Algorithms and Applications , pp. 101–122. Springer. [8] Pozzi A, Xie X, Raimondo DM, Schenkendorf R. 2020 Global sensitivity methods for design of experiments in lithium-ion battery context. IFAC-PapersOnLine 53, 7248–7255. [9] Chu Y, Hahn J. 2013 Necessary condition for applying experimental design criteria to global sensitivity analysis results. Computers & Chemical Engineering 48, 280–292. [10] Lambert B, Lei CL, Robinson M, Clerx M, Creswell R, Ghosh S, Tavener S, Gavaghan DJ. 2023 Autocorrelated measurement processes and inference for ordinary differential equation models of biological systems. Journal of the Royal Society Interface 20, 20220725. [11] Fu X, Patel HP, Coppola S,	https://arxiv.org/abs/2504.19233v1
Xu L, Cao Z, Lenstra TL, Grima R. 2022 Quantifying how post-transcriptional noise and gene copy number variation bias transcriptional parameter inference from mRNA distributions. eLife 11, e82493. [12] Kuhlmann H. 2001 Importance of autocorrelation for parameter estimation in regression models. In Proceedings of the 10th FIG International Symposium on Deformation Measure- ments, Orange, California . Citeseer. [13] Simoen E, Papadimitriou C, Lombaert G. 2013 On prediction error correlation in Bayesian model updating. Journal of Sound and Vibration 332, 4136–4152. 22 [14] Durbin J, Koopman SJ. 2012 Time series analysis by state space methods vol. 38 Oxford Statistical Science Series . Oxford Academic 2nd edition. [15] Biagini F, Hu Y, Øksendal B, Zhang T. 2008 Stochastic Calculus for Fractional Brownian motion and Applications . Springer Science & Business Media. [16] Dereich S, Neuenkirch A, Szpruch L. 2012 An Euler-type method for the strong approxima- tion of the Cox–Ingersoll–Ross process. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, 1105–1115. [17] Maller RA, M¨ uller G, Szimayer A. 2009 Ornstein–Uhlenbeck processes and extensions. Handbook of Financial Time Series pp. 421–437. [18] Borgonovo E, Plischke E. 2016 Sensitivity analysis: A review of recent advances. European Journal of Operational Research 248, 869–887. [19] Saltelli A, Sobol’ IM. 1995 Sensitivity analysis for nonlinear mathematical models: numer- ical experience. Matematicheskoe Modelirovanie 7, 16–28. [20] Cannav´ o F. 2012 Sensitivity analysis for volcanic source modeling quality assessment and model selection. Computers & Geosciences 44, 52–59. [21] Banks HT, Holm K, Kappel F. 2011 Comparison of optimal design methods in inverse problems. Inverse Problems 27, 075002. [22] Bauer I, Bock HG, K¨ orkel S, Schl¨ oder JP. 2000 Numerical methods for optimum experi- mental design in DAE systems. Journal of Computational and Applied Mathematics 120, 1–25. [23] Danilova M, Dvurechensky P, Gasnikov A, Gorbunov E, Guminov S, Kamzolov D, Shibaev I. 2022 Recent theoretical advances in non-convex optimization. In High-Dimensional Op- timization and Probability: With a View Towards Data Science , pp. 79–163. Springer. [24] Tian Y, Cheng R, Zhang X, Jin Y. 2017 PlatEMO: A MATLAB platform for evolutionary multi-objective optimization. IEEE Computational Intelligence Magazine 12, 73–87. [25] Tian Y, Zhu W, Zhang X, Jin Y. 2023 A practical tutorial on solving optimization problems via PlatEMO. Neurocomputing 518, 190–205. [26] Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingm¨ uller U, Timmer J. 2009 Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25, 1923–1929. 23 [27] Royston P. 2007 Profile likelihood for estimation and confidence intervals. The Stata Journal 7, 376–387. [28] Simpson MJ, Baker RE. 2024 Parameter identifiability, parameter estimation and model prediction for differential equation models. arXiv preprint arXiv:2405.08177 . 24 Supplementary Information Optimal experimental design for parameter estimation in the presence of autocorrelated observation noise Jie Qi1and Ruth E. Baker2 1College of Information Science and Technology, Donghua University, Shanghai, China 2Mathematical Institute, University of Oxford, Oxford, United Kingdom 1arXiv:2504.19233v1 [math.ST] 27 Apr 2025 S1 Parameter identifiability under different noise models Figure S1 shows equivalent results to those shown in Figure 1 of	https://arxiv.org/abs/2504.19233v1
the main text, but with decreased noise correlation ( ϕ= 0.1 compared to ϕ= 0.02 in the main text). Figure S1: Parameter identifiability under the different noise models, with data collected at 11 regularly spaced time points t= 0,8,16, . . . , 80. The left column shows the sampled data, the underlying model solution using the input parameters, and the solution generated using the MLE parameters. The shaded region shows the 95% prediction interval generated using the 95% confidence intervals from the profile likelihoods. The right-most three panels show the profile likelihoods for each parameter. The results in the top row are generated using uncorrelated Gaussian noise with σ2 IID= 9.0. The results in the middle row are generated using correlated OU noise with ϕ= 0.1 and σ2 OU/(2ϕ) = 9.0. The results in the bottom row are generated using a misspecified noise model: the data are generated using correlated OU noise with ϕ= 0.1 and σ2 OU/(2ϕ) = 9.0, whilst the analysis is carried out assuming IID Gaussian noise with σ2 IID= 9.0. 2 S2 Confidence interval widths for autocorrelated noise We perform parameter inference using measurement times optimised under both IID and OU noise assumptions, where the actual measurement noise follows an OU process. The resulting mean parameter confidence interval widths are shown in Figure S2, illustrating the impact of different autocorrelation levels in the OU noise. Figure S2: Mean confidence interval widths generated using 1000 simulations, under OU noise with σ2 OU/(2ϕ) = 4.0 and (from top to bottom row) ϕ= 0.02,0.12,0.30. We perform parameter inference using the measurement times optimised with the FIM and Sobol indices under both IID and OU noise assumptions. 3 S3 Sensitivity of the confidence intervals to small changes in measurement times We assessed the change in confidence intervals with small changes in the measurement times, as per the following matrix: t= 0 20 40 60 80 0 20 40 60 64 80 0 13 .3 20 40 60 80 0 11 .4 20 40 45 .7 60 68 .6 80 0 10 20 30 40 50 60 70 80 . (S1) The results are shown in Figure S3, and demonstrate that small changes in the measurement points can significantly affect the confidence intervals for the growth rate, r. Figure S3: Variation of the confidence interval width as the placement of the measurement points is varied according to Equation (S1). In each case, we plot the mean confidence interval as a function of the number of observation points, ns. The plots in each row are generated using 1000 simulations and σ2 IID= 0.64. 4	https://arxiv.org/abs/2504.19233v1
arXiv:2504.19331v1 [math.ST] 27 Apr 2025Bahadur asymptotic eﬃciency in the zone of moderate deviation probabilities Mikhail Ermakov April 2025 Institute of Problems of Mechanical Engineering RAS,and St . Petersburg State University Аннотация For a sequence of independent identically distributed rand om variables having a distribution function with an unknown parameter fr om a setΘ⊂Rd, we prove an analogue of the lower bound of Bahadur asymptotic eﬃciency for the zone of moderate deviation prob abilities. The assumptions coincide with assumptions conditions unde r which the locally asymptotically minimax lower bound of Hajek-Le Cam was proved. The lower bound for local Bahadur asymptotic eﬃcienc y is a special case of this lower bound. 1 Introduction LetX1,...,X nbe independent indentically distributed random variables having probability measure Pθ,θ∈Θ. Probabilitu measures Pθ,θ∈Θ, are deﬁned on a σ- algebra Bof setS. The set Θis open bounded subset ofRd. The value of parameter θis unknown. We are interested in the lower bounds of asymptotic eﬃciency for estimators of parameter θ. There are two approaches for the study of asymptotic eﬃcienc y. One of them is a locally asymptotically minimax lower bound. Local ly asymptotically minimax lower bound of the Hajek-Le Cam [7, 9, 8, 11] speciﬁes a lower bound 1The research was supported by the Ministry of Science and Hig her Education of Russian Federation (project 124041500008-1). 1 on the asymptotic eﬃciency for estimators ˆθn=ˆθn(X1,...,X n)that deviate from the true value of the parameter θby an order of n−1/2. In this setting we get the lower bound of the supremum of risk of statistical e stimates for any arbitrarily small neighborhood of the true value of the p arameter. Such a lower bound of asymptotic eﬃciency does not provide precis e information about the behavior of the risk of statistical estimator for a speciﬁc value of the parameter. The lower bound for the Bahadur asymptotic eﬃciency [1, 2, 3, 8], proved for probabilities of large deviations of statistical estim ates, provides the lower bounds of the risks of statistical estimates at each speciﬁc point of possible parameter values. In this case, only consistency of estimat es is assumed. In the zone of moderate deviation probabilities bounds for l ocally asymptotically minimax risk of statistical estimators have been establish ed in [4, 5, 10]. The interest in this issue is due to the fact that the constructin g the boundaries of conﬁdence intervals is based on tails of estimator distribu tions and, therefore, can be treated as a problem of moderate deviation probabilit ies. For logarithmic asymptotics, locally asymptotically minimax lower bounds for the moderate deviation probabilities were obtained under the same assum ptions [4, 10] as the lower bound for the local asymptotic eﬃciency of Hajek -Le Cam [7, 9, 8, 11]. For the strong asymptotics of moderate deviati on probabilities of statistical estimators, the lower bound of the asymptoti cally minimax risk has been proved under not very strong additional assumption s [4, 5]. In the zone of moderate deviation probabilities, one can pro ve lower bounds for asymptotic eﬃciency both in the asymptotically m inimax setting and in	https://arxiv.org/abs/2504.19331v1
the Bahadur setting. For the asymptotically minimax s etting the lower bounds have been proved in [4, 5, 10]. The goal of presen t paper is to obtain an analogue of the Bahadur lower bound in the zone of moderate deviation probabilities assuming the same conditions as th e lower bound on the Hajek-Le Cam local asymptotic eﬃciency. Note that the st raightforward application of Bahadur method of the proof leads to signiﬁca nt additional conditions [6, 8] even for establishing the lower bound for t he local Bahadur asymptotic eﬃciency. The lower bound for the local Bahadur a symptotic eﬃciency is a special case of the lower bound on the asymptoti c eﬃciency in the zone of moderate deviation probabilities established i n the present paper. ForΘ⊂R1, we also prove an analogue of Bahadur lower bound separately on each side of the exterior of the interval to which the estim ator should not belong. A multidimensional analogue of these “one-sided” l ower bounds was also obtained. 2 The paper is organized as follows. All results and proofs are collected in section 2. In subsection 2.1 we introduce the condition un der which the main results are proved. This condition is the same one under which the locally asymptotically minimax Hajek-Le Cam risk bound is e stablished. In subsection 2.2, for the sake of completeness of presentat ion and better understanding of further results, a locally asymptoticall y minimax risk bound for moderate deviation probabilities of statistical estim ates is presented. In subsection 2.3, analogue of the lower bound for Bahadur asym ptotic eﬃciency of statistical estimators is provided in the zone of moderat e deviation probabilities. The lower bounds for Bahadur local asymptotic eﬃciency are p articular cases of these ones. In subsection 2.4 we give a generalization of B ahadur lower bounds to the multivariate case, covering "one-sided"lowe r bounds, and show that its proof is no diﬀerent from the proof of traditional Ba hadur lower bounds on the asymptotic eﬃciency [3, 8]. For the case of multidimensional parameter θ∈Θ⊂Rd,d >1, we shall denote by Greek letters θ,τ,φ,... vectors or vector functions. We denote 1(A)the indicator of an event A. 2 Main Results 2.1 Main condition Suppose probability measures Pθ,θ∈Θ, аre absolutely continuous with respect to probability measure ν, deﬁned on the same σ- algebra Bof setS, and have the densities f(x,θ) =dPθ dν(x), x∈S. (2.1) For any θ,θ0∈Θ, denote Pa θθ0andPs θθ0absolutely continuous and singular components of probability measure Pθwith respect to probability measure Pθ0. For allθ0,θ0+τ∈Θdeﬁne the function g(x,τ) =g(x,θ0,θ0+τ) =/parenleftBigf(x,θ0+τ) f(x,θ0)/parenrightBig1/2 −1 (2.2) for allxbelonging to the support of measure Pa θ0+τ,θ0and equal to zero otherwise. 3 We say that statistical experiment E={(S,B),Pθ,θ∈Θ}has the ﬁnite Fisher information at the point θ0∈Θ, if there exists vector function φ: S→Rdsuch that we have /integraldisplay S(g(x,τ)−τTφ)2dPθ0=o(\|τ\|2),Ps θ0,θ0+τ(S) =o(\|τ\|2) (2.3) asτ→0and the matrix components I(θ0) = 4/integraldisplay SφφTdPθ0. (2.4) take on ﬁnite values. MatrixI(θ0)is called Fisher information matrix. Let we have a sequence un>0,un→0,nu2 n→ ∞ asn→ ∞. We say that estimator ˆθn=θn(X1,...,X n)of parameter θ∈Θisun- consistent, if for any θ0∈Θthere	https://arxiv.org/abs/2504.19331v1
is a vicinity Uofθ, such that, for any δ >0, we have lim n→∞sup θ∈UPθ(\|ˆθn−θ\|> δun) = 0. (2.5) 2.2 Locally asymptotically minimax lower bound The locally asymptotically minimax lower bound of risks in t he zone of moderate deviation probabilities does not require any cond itions of consistency. Theorem 2.1 Let the statistical experiment E={(S,B),Pθ,θ∈Θ}have the ﬁnite Fisher information at the point θ0∈Θ. Then, for any estimator ˆθn, for the points θ0,θn=θ0+2un∈Θ, we have lim inf n→∞sup θ=θ0,θn(nu2 nI(θ)/2)−1logPθ(\|ˆθn−θ\|> un)≥ −1.(2.6) For the proof we consider the problem of hypothesis testing H0:θ=θ0 versus alternative Hn:θ=θ0+vn,vn= 2un. Deﬁne the test Kn= Kn(X1,...,X n) =1(ˆθn−θ0> un). Denote α(Kn)andβ(Kn)respectively type I and type II error probabilities of test Kn. By Theorem 2.2 in [4], if α(Kn)< c <1andβ(Kn)< c <1, we have lim sup n→∞(nv2 n)−1/2(\|2logα(Kn)\|1/2+\|2logβ(Kn)\|1/2)≤1 (2.7) 4 Proof of (2.7) is based on the extension of a certain version o f the statement about local asymptotic normality to the zone of probabiliti es of moderate deviations and the fact that (2.7) is valid in the case of test ing a similar hypothesis for the normal distribution by virtue of the Neym an-Pearson Lemma. We have α(Kn)≤Pθ0(\|θn−θ0\|> un) (2.8) and β(Kn) =Pθn(ˆθn−θ0< un) =Pθn(ˆθn−θ0−2un<−un) =Pθn(\|ˆθn−θn\|> un). (2.9) By (2.7)- (2.9), we get (2.6). 2.3 Bahadur eﬃciency in the zone of moderate deviation probabilities Let us present an analogue of lower bound for Bahadur eﬃcienc y in the zone of moderate deviation probabilities d= 1. Theorem 2.2 Let the statistical experiment E={(S,B),Pθ,θ∈Θ}has the ﬁnite Fisher information at the point θ0∈Θ⊂R1. Let estimator ˆθnbeun- consistent. Then, for any estimator ˆθn, we have lim inf n→∞(nu2 nI(θ)/2)−1logPθ(\|ˆθn−θ\|> un)≥ −1. (2.10) Moreover, we have lim inf n→∞(nu2 nI(θ)/2)−1logPθ(ˆθn−θ > un)≥ −1. (2.11) For the proof we consider the problem of hypothesis testing H0:θ=θ0 versus alternatives Hn:θ=θ0+vn,vn=run,r >1. Deﬁne the tests Kn=Kn(X1,...,X n) =1(\|ˆθ−θ0\|> un). Since the estimator ˆθnisun-consistent, we can implement (2.7) and we get lim sup n→∞(nv2 n)−1/2(\|2logPθ0(\|ˆθn−θ0\|> un)\|1/2+\|2logPθ0+vn(\|ˆθn−θ0\|< un)\|1/2)≤1. (2.12) 5 Since the estimator ˆθnisun-consistent, then lim n→∞Pθ0+vn(\|ˆθn−θ0\|< un) = 0. (2.13) By (2.19) and (2.13), we get liminf sup n→∞(nr2u2 n)−1/2(\|2logPθ0(\|ˆθn−θ0\|> un)\|1/2≤1. (2.14) Since the choice of r >1is arbitrary, we get (2.10). Inequality (2.11) is proved similarly. It suﬃces to impleme nt (2.7) to the testKn=1(ˆθn−θ0> un). This reasoning are omitted. The multidimensional version of Theorem 2.2 is provided bel ow. In the following Theorems 2.3 and 2.5 the asymptotic eﬃciency vari es in diﬀerent directions. This is the main distinguishing feature of Theo rems. LetVbe bounded convex open set in Rdand0∈V. Denote ¯Vthe complement of set V. Theorem 2.3 Let the statistical experiment E={(S,B),Pθ,θ∈Θ}has the ﬁnite Fisher information at the point θ0∈Θ⊂Rd. Let estimator ˆθnbeun- consistent. Then, for any estimator ˆθn, we have lim inf n→∞(nu2 n)−1logPθ(ˆθn−θ∈un¯V)≥ −1 2inf τ∈¯VτTI(θ)τ (2.15) Moreover, for any cone Kwith a vertex at zero and having a non-empty interior, there is lim inf n→∞(nu2 n)−1logPθ(ˆθn−θ∈un¯V∩K)≥ −1 2inf τ∈¯V∩KτTI(θ)τ.(2.16) The case of θ∈Θ⊂Rdis similar to θ∈Θ⊂R1and also reduces to the problem of estimating a parameter at two points. The following lower bound for local asymptotic Bahadur eﬃci ency	https://arxiv.org/abs/2504.19331v1
is deduced from Theorem 2.3. . We call estimator ˆθnlocally uniformly consistent, if for any θ0∈Θ, there is a neighborhood Uof the point θ0such that, for anyε >0, there is lim n→∞sup θ∈UPθ(\|ˆθn−θ\|> ε) = 0. (2.17) 6 Theorem 2.4 Let the statistical experiment E={(S,B),Pθ,θ∈Θ}has the ﬁnite Fisher information at the point θ∈Θ⊂R1. Let estimator ˆθnbe locally uniformly consistent. Then, for any estimator ˆθn, we have . lim inf u→0lim n→∞(nu2)−1logPθ(ˆθn−θ∈u¯V)≥ −1 2inf τ∈¯VτTI(θ)τ. (2.18) Moreover, for any cone Kwith a vertex at zero and having a nonempty interior, there is lim inf u→0lim n→∞(nu2)−1logPθ(ˆθn−θ∈u¯V∩K)≥ −1 2inf τ∈¯V∩KτTI(θ)τ.(2.19) The requirement of local uniform consistency replaces the c onsistency requirement under which the lower bound of Bahadur asymptotic eﬃciency i s usually proved. However, under the consistency condition, the proo f of (2.16) requires the introduction of additional regularity conditions for t he family of probability measures Pθ,θ∈Θ(Th. 9.3, Ch.1 in [8] and [6]). Note that the requirement (2.3) of diﬀerentiability of the f unctionginL2 can be replaced by weaker conditions (2.5)-(2.7) in [4] on th e behavior of the function gitself. 2.4 Multidimensional lower bound of Bahadur asymptotic eﬃciency Estimator ˆθnis called the consistent estimator of parameter θ∈Θ, if for any θ∈Θ, for any ε >0, we have lim n→∞Pθ(\|ˆθn−θ\|> ε) = 0. (2.20) Theorem 2.5 Letˆθnbe a consistent estimator of parameter θ∈Θ. Let θ0∈Θ. LetΩ⊂Rdbe open set such that 0/∈Ωandθ0+Ω⊂Θ. Then, for any ˜θ∈θ0+Ω, we have lim n→∞1 nlogPθ0(ˆθn−θ0∈Ω)≥ −/integraldisplay Slogf(x,˜θ) f(x,θ0)f(x,˜θ)ν(dx).(2.21) Proof is akin to [3, 8]. Denote −Kthe righthand side of (2.21). By Jensen inequality, the righthand side of (2.21) is not positive. 7 We deﬁne the indicator function λn=λn(ˆθn−θ0) = 1, ifˆθn−θ0∈Ωand λn=λn(ˆθn−θ0) = 0, ifˆθn−θ0/∈Ω. We putr=n(K+δ)withδ >0. Denote Gn=Gn(X1,...,X n,θ0,˜θ) =n/productdisplay j=1f(Xj,˜θ) f(Xj,θ0). (2.22) We have Pθ0(ˆθn−θ0∈Ω) =Eθ0λn ≥Eθ0(λn1(Gn<exp{r}))≥exp{−r}E˜θ/braceleftBig λn1(Gn<exp{r})/bracerightBig ≥exp{−r}(P˜θ(ˆθn−θ0∈Ω)−P˜θ(Gn>exp{r})).(2.23) Sinceˆθnis consistent estimator, we have lim n→∞P˜θ(ˆθn−θ0∈Ω) (2.24) By the Law of Large Numbers, we have lim n→∞P˜θ/parenleftBig1 n\|Gn−nK\|> δ/2/parenrightBig = 0. (2.25) By (2.23) -2.25), we get (2.21). Список литературы [1] Bahadur, R.R. (1960). Asymptotic eﬃciency of tests and e stimates. Sankhya. 22 229–252. [2] Bahadur, R.R. Rates of convergence of estimates and test statistics. Ann.Math.Stat. 38(1967) 303-324 [3] Bahadur, R.R., Gupta J.C. and Zabel S.L. Large deviation s of tests and estimates. In Asymptotic Theoryof Statistical Tests and Es timation.(ed. I.M. Chakravati) (1980) 33-64 Academic NY [4] Ermakov, M.S. (2003). Asymptotically eﬃcient statisti cal inference for moderate deviation probabilities. Theory Probab. Appl. 48 (4) 676–700. 8 [5] Ermakov, M.S. The sharp lower bounds of asymptotic eﬃcie ncy of estimators in the zone of moderate deviation probabilities . Electronic Journal of Statistics 6(2012) 2150-2184. [6] Fu J.C. On a theorem of Bahadur on rate of convergence of po int estimators. Ann.Stat. 4(1973)745-749) [7] Hajek, J. (1972). Local asymptotic minimax and admisibi lity in estimation. Proc.Sixth Berkeley Symp. on Math. Statist. an d Probab. 1 175–194, Berkeley, California Univ. Press [8] Ibragimov, I.A. and Hasminskii, R.Z. (1981). Statistic al Estimation: Asymptotic Theory. Berlin: Springer. [9] Le Cam, L. (1972). Limits of Experiments Berlin: Springe r. Proc.Sixth Berkeley Symp. on Math. Statist. and Probab. 1 245–261,	https://arxiv.org/abs/2504.19331v1
arXiv:2504.19337v1 [math.ST] 27 Apr 2025Submitted to Bernoulli Frequency Domain Resampling for Gridded Spatial Data SOUVICK BERA1,a, DANIEL J. NORDMAN2,cand SOUTIR BANDYOPADHYAY1,b 1Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401, USA, aberasouvick@mines.edu ,bsbandyopadhyay@mines.edu 2Department of Statistics, Iowa State University, Ames, IA 50011, USA,cdnordman@iastate.edu In frequency domain analysis for spatial data, spectral averages based on the periodogram often play an impor- tant role in understanding spatial covariance structure, but also have complicated sampling distributions due to complex variances from aggregated periodograms. In order to nonparametrically approximate these sampling dis- tributions for purposes of inference, resampling can be useful, but previous developments in spatial bootstrap have faced challenges in the scope of their validity, speciﬁcally due to issues in capturing the complex variances of spatial spectral averages. As a consequence, existing frequency domain bootstraps for spatial data are highly restricted in application to only special processes (e.g. Gaussian) or certain spatial statistics. To address this lim- itation and to approximate a wide range of spatial spectral averages, we propose a practical hybrid-resampling approach that combines two different resampling techniques in the forms of spatial subsampling and spatial boot- strap. Subsampling helps to capture the variance of spectral averages while bootstrap captures the distributional shape. The hybrid resampling procedure can then accurately quantify uncertainty in spectral inference under mild spatial assumptions. Moreover, compared to the more studied time series setting, this work ﬁlls a gap in the theory of subsampling/bootstrap for spatial data regarding spectral average statistics. Keywords: Spatial Frequency Domain Bootstrap; Periodogram; Spectral Mean 1. Introduction In recent years, there has been a marked increase in research on the analysis of spatial data using the fre- quency domain approach (see, for example, ( Bandyopadhyay, Jentsch and Rao ,2017,Bandyopadhyay and Lahiri ,2010 ,Bandyopadhyay, Lahiri and Nordman ,2015 ,Bandyopadhyay and Rao ,2016 ,Fuentes , 2006 ,2007 ,Hall, Fisher and Hoffman ,1994 ,Im, Stein and Zhu ,2007 ,Matsuda and Yajima ,2009 , Van Hala et al. ,2017,2020 )). As a beneﬁt, analysis in the frequency domain allows inference about covariance structures through a data transformation (i.e., a Fourier or periodogram-based transform), without the need for a full probability model for spatial data. Resampling methods, such as subsam- pling and bootstrap, for spatial data have gained popularity in recent decades because these approaches can often allow for uncertainty quantiﬁcation and distributional approximations for complicated spa- tial statistics in an nonparametric or model-free fashion; see ( Davison and Hinkley ,1997 ,Sherman and Carlstein ,1994 ,Sherman ,1998 ). In a recent study, ( Ng, Yau and Chen ,2021 ) introduced a type of frequency domain bootstrap (FDB) for Gaussian spatial data on grid. Although this work offers a signiﬁcant contribution to nonparametric spatial approximations, the resampling methodology comes with certain hard limitations in application. Firstly, the FDB of ( Ng, Yau and Chen ,2021 ) is generally not valid for approximating many spatial spectral averages of interest, because the latter have complex variance structures that this FDB proposal cannot correctly estimate. In particular, the challenge is that spectral mean statistics have variances that depend both on	https://arxiv.org/abs/2504.19337v1
spatial covariances as well as higher or- der (i.e., fourth) process cumulants, which arise due to covariances between the spatial periodogram ordinates. Technically, it is the variance contributions related to higher order cumulants that are gener- ally missed or ignored in the FDB approach of ( Ng, Yau and Chen ,2021 ), unless the spatial data are 1 2 Gaussian (i.e., so that these variance components become zero). Consequently and secondly, the spatial bootstrap results of ( Ng, Yau and Chen ,2021 ) are established under assumptions of Gaussianity, which can be stringent and may not suitably reﬂect the distributional nature of spectral average statistics in general practice. These aspects motivate a need to modify the spatial FDB of ( Ng, Yau and Chen , 2021 ) to handle a broader range of spectral statistics with application potentially non-Gaussian spatial processes. To provide improved and more universally valid distributional approximations with spatial spec- tral statistics, we introduce a resampling technique called spatial Hybrid Frequency Domain Bootstrap (HFDB) which aims to merge two different resampling schemes in the forms of subsampling and boot- strap into one overall approach in the frequency domain. The idea that spatial subsampling can be used to correctly estimate the variances of complicated spectral averages, while bootstrap can be used to re-create the shape of the sampling distributions. Essentially then, subsampling plays a role in ap- propriately modifying the spreads of FDB distributional approximations, where this bootstrap would otherwise be invalid without such scaling adjustments. The advantage of the proposed HFDB method is that this can accurately capture the uncertainty in spatial spectral statistics for calibrating tests or conﬁdence intervals without requiring stringent conditions on the spatial processes or assumptions of Gaussianity; in particular, the conditions needed to validate HFDB basically amount to mild assump- tions for ensuring that spectral averages have limit distributions at all. This work then intends to close some gaps in the scope of applying resampling approximations for spatial data, which is practically valuable. Additionally, our work also bridges some gaps in the theory of subsampling and bootstrap for spatial data, and extends some recent theoretical ﬁndings for spectral inference with time series to the spatial data setting. We conclude this section by brieﬂy overviewing FDB for time series and providing connections to the spatial bootstrap here. In time series analysis, FDB has received much attention and interest to- ward approximating the distributions of spectral statistics derived from the periodogram (cf. ( Kreiss and Paparoditis ,2012 ,Kreiss and Lahiri ,2012 ,Lahiri ,2003 ,Politis and McElroy ,2019 )). The key con- cept behind this method is to mimic a type of asymptotic independence that exists in the periodogram in order to generate versions of periodogram ordinates in the bootstrap world by independent resam- pling (cf. ( Dahlhaus and Janas ,1996 ,Jentsch and Kreiss ,2010 ,Kreiss and Paparoditis ,2003 ,2012 , 2023 ,Kreiss, Paparoditis and Politis ,2011)). In the last three decades, there has been a progression of FDB methods, often relying on speciﬁc assumptions about the underlying time series (e.g., linear process) or differing cases of	https://arxiv.org/abs/2504.19337v1
spectral mean parameters. Recently, a major advancement was achieved by (Meyer, Paparoditis and Kreiss ,2020 ) who introduced an innovative bootstrap method, called the hybrid periodogram bootstrap (HPB), that represents the state-of-the-art approach for time series in- ference concerning spectral means. Related to this, ( Yu, Kaiser and Nordman ,2023 ) recently studied subsampling for use in conjunction with HPB, where subsampling helps to justify the latter bootstrap approach under weaker assumptions than ( Meyer, Paparoditis and Kreiss ,2020 ) and so HPB extends to a wide range of application with various time processes. Unlike bootstraps that aim to re-create statis- tics at the same data level as the original sample, subsampling is a different resampling approach that generates smaller-scale copies of statistic; because of this distinction, subsampling often applies under weaker conditions than bootstrap, though bootstrap can provide more accurate distributional approxi- mations when valid (cf. ( Politis, Romano and Wolf ,1999 )). The spatial FDB proposed by ( Ng, Yau and Chen ,2021 ) is the spatial equivalent of one of the original FBD methods for time series (cf. ( Dahlhaus and Janas ,1996 ,Jentsch and Kreiss ,2010 ,Kreiss and Paparoditis ,2003 )), while the proposed spatial HFDB here intends to be a spatial analog of the HPB for time series as studied in ( Meyer, Paparodi- tis and Kreiss ,2020 ) when combined with a subsampling extension as in ( Yu, Kaiser and Nordman , 2023 ). This development is non-trivial for spatial data and, similar to the time series setting, is crucial for avoiding restrictive process and moment conditions for dependent data. Finally, it is worth men- tioning that, similar to ( Ng, Yau and Chen ,2021 ), we focus our resampling presentation on spatial Spatial Frequency Domain Bootstrap 3 observations lying on a lattice in two-dimensional space, though ﬁndings can be extended to gridded data in more general spatial sampling dimensions Ěg2. The remainder of the paper is organized as follows. Section 2introduces the distributions of spec- tral mean statistics, which form the basis for inference, along with the associated framework. Section 3presents a spatial subsampling framework in the frequency domain and establishes its consistency, forming the foundation for the proposed bootstrap procedure. A detailed discussion of the hybrid boot- strap method for spatial data is provided in Section 4. Section 5presents numerical studies to evaluate the accuracy of the proposed procedure, and Section 6demonstrates an application for calibrating spatial isotropy tests. Finally, Section 7concludes with key insights. Additional technical details and simulation results can be found in Appendix andSupplementary Material , respectively. 2. Distributions of Spectral Mean Statistics 2.1. Spatial process and sampling design Throughout this paper, we follow spatial sampling framework similar to ( Bandyopadhyay and Lahiri , 2010 ) and ( Ng, Yau and Chen ,2021 ). Let{Ė(s)}:s∈Z2}denote a real-valued second-order stationary process located on a regular integer grid Z2. In this fashion, potential observations lie on a spatial lattice with a constant separation in each coordinate direction (which we take to be 1 though other scaling is possible as well). We assume that	https://arxiv.org/abs/2504.19337v1
the random process Ė(s)is observed at Ĥ≡Ĥ1Ĥ2sampling sites deﬁned by {s1,...., sĤ}={s∈Z2:s∈D(Ĥ1×Ĥ2)}=D(Ĥ1×Ĥ2)∩Z2 given by those locations on the grid Z2that lie within a rectangular sampling region D(Ĥ1×Ĥ2) ≡ [1,Ĥ1] × [1,Ĥ2]for integersĤ1,Ĥ2g1. For spatial data analysis, the above sampling framework corresponds to a so-called pure increasing domain for developing spatial results (cf. ( Cressie ,1993 )). That is, in this scheme, more spatial obser- vations are available (i.e., Ĥ→ ∞ ) as spatial sampling region expands in size in each direction (i.e., Ĥğ→∞ forğ=1,2). 2.2. Spectral mean parameters Suppose that the second order stationary spatial process {Ė(s):s∈Z2}has a spectral density denoted asĜ(ā):Π2−→ [ 0,∞)whereĜ(ā) ≡ ( 2ÿ)−2/summationtext.1 h∈Z2Cov(Ė(0),Ė(h))exp(−ăhTā), foră≡√ −1 and forā∈Π2≡ [−ÿ,ÿ]2. Then, the target spatial parameter for inference is a spectral mean parameter deﬁned as an integral ĉ(ć)=/uni222B.dsp Π2ć(ā)Ĝ(ā)Ěā, (1) involvingĜand some chosen/given function ć:Π2−→Rof bounded variation. Spectral mean param- eters are common in the spatial data analysis for studying the spatial covariance structure of the data with an unknown distribution. Such quantities have a well-established history for both time series and spatial data (see, ( Politis and McElroy ,2019 ) for more details), dating as far back as ( Parzen ,1957 ), and many spatial parameters of interest can be expressed in the form of a spectral mean. Standard examples of spectral means include autocovariances and spectral distributions, as described below, depending on the choice of ćabove. Next we give some examples of spectral mean parameters arising in frequency 4 domain analysis (cf. ( Bandyopadhyay, Lahiri and Nordman ,2015 ,Dahlhaus ,1985 ,Dahlhaus and Janas , 1996 ,Parzen ,1957 ,Subba Rao ,2018 ,Van Hala et al. ,2017,2020 ) and the references therein). Examples of spectral mean parameters 1. Covariance function. For a lag vector h∈Z2, if we consider the function ć(ā)=cos(hTā)in1, this leads to the autocovariance function Ā(h)=Cov(Ė(0),Ė(h)), which can be viewed as a spectral mean ĉ(ć). Note that, for testing the special hypothesis Ą0:Ė(s)is white noise (constant Ĝ), a test based on the process autovariances can be applied, similar to some Portmanteau tests (cf. ( Li and McLeod , 1986 ,Ljung and Box ,1978 )) using the fact that the spectral means ĉ(ć)are zero under this Ą0for the function choice ć(ā) ≡ [cos(h1Tā),...,(cos(hġTā)]Tfor some integer lags h1,..., hġ. 2. Spectral distribution function. For a vector t∈R2, the spectral distribution function is deﬁned as Ă(t)=/uni222B R2I(−∞,t](ā)Ĝ(ā)Ěā, which corresponds to a spectral mean parameter deﬁned by Ā(h)= I(−∞,t](ā)in (1) whereI(·)is the indicator function and (−∞,t] ≡ (−∞,Ī1] × (−∞,Ī2]. The spectral distribution function plays an important role in determining the smoothness of the sample paths of the spatial process Ė(·)(cf. ( Stein ,1999 )). 3. Assessment of spatial covariance structures. When analyzing spatial data, it can be useful to assess the spatial covariance structures in a nonparametric manner, without requiring speciﬁc distributional as- sumptions about the data. General testing methods can be developed of evaluating different hypotheses about spatial covariances, such as tests of isotropy or separability, by examining ĉ(ć)with appropriate choices ofć(ā)functions. For more details, refer to ( Van Hala et al. ,2017,2020 ). 4. Whittle Estimation. Suppose{ĜĂ}represents a parametric	https://arxiv.org/abs/2504.19337v1
family of spectral densities (indexed by Ă). Whittle estimation aims to identify the closest member ĜĂto the true density Ĝ(cf. ( Taniguchi ,1979 )). Assuming real-valued Ăfor illustration, the solution to a spectral mean equation ĉ(ć)=0 identiﬁes the appropriate ĜĂunder certain conditions, using a function ć(ā) ≡ [ 1−ĜĂ(ā)/Ĝ(ā)]Ě ĚĂĜ−1 Ă(ā),ā∈ R2, deﬁned by the derivative of Ĝ−1 Ă(ā) ≡1/ĜĂ(ā). 5. Goodness-of-ﬁt tests. There has been increasing interest in frequency domain based tests to assess model adequacy (cf. ( Crujeiras and Fernandez-Casal ,2010 ,Van Hala et al. ,2020 ,Weller and Hoeting , 2020 )). Consider a test involving a simple null hypothesis Ą0:Ĝ=Ĝ0against an alternative Ą0:Ĝ≠Ĝ0 for some candidate spectral density Ĝ0. One immediate test for Ą0is based on the function ć(ā)= 1/Ĝ0(ā)withĉ(ć)a constant under this simple Ą0. To test the composite hypothesis Ą0:Ĝ∈ F for a speciﬁed parametric class F, several frequency domain tests have been proposed in time series (cf. ( Beran ,1992 ), (Milhøj ,1981 ), (Paparoditis ,2000 ), (Nordman and Lahiri ,2006 )). These tests can use Whittle estimation to choose the “best” ﬁtting model from Fand then compare the ﬁtted density to the periodogram across all ordinates. One can adapt strategies for goodness-of-ﬁt tests with time series (cf. ( Nordman and Lahiri ,2006 )), which combine aspects of model ﬁtting and model comparison into a choice of estimating function ć, to spatial processes. 6. Variogram model ﬁtting. A popular approach to ﬁtting a parametric variogram model to spatial data is through the method of least squares; see ( Cressie ,1993 ). Let{2Ā(·;Ă):Ă∈Θ},Θ¢RĦdenote a class of valid variogram models for the true variogram 2 Ā(h) ≡Var(Ė(h) −Ė(0)),h∈Z2of the spa- tial process. Let 2 /hatwideĀĤ(h)denote the sample variogram at lag hbased onĖ(s1),...,Ė(sĤ)(cf. Chapter 2, (Cressie ,1993 )). Then one can ﬁt the variogram model by estimating the parameter Ăthat minimizes the criterion: /hatwideĂĤ=argmin/braceleftBig/summationtext.1ģ ğ=1/parenleftBig 2/hatwideĀĤ(hğ) −2Ā(hğ;Ă)/parenrightBig2 :Ă∈Θ/bracerightBig for a given set of lags h1,..., hģ. Expressing the variogram in terms of the spectral density function, we get an equivalent spectral esti- Spatial Frequency Domain Bootstrap 5 mating equation for identifying Ă:ĉ(ć)=0 for ĉ(ć) ≡/uni222B.dsp R2/bracketleftBigģ/summationdisplay.1 ğ=1/braceleftBig 1−cos(hğTā)−Ā(h;Ă)/bracerightBig ∇[2Ā(hğ;Ă)]/bracketrightBig Ĝ(ā)Ěā, usingćĂ(ā) ≡/summationtext.1ģ ğ=1/braceleftBig 1−cos(hğTā)−Ā(h;Ă)/bracerightBig ∇[2Ā(hğ;Ă)]. 2.3. Spectral mean statistics and sampling distributions Based on the stationary spatial process Ė(·)as observed on the sampling region D(Ĥ1×Ĥ2), where Ĥ=Ĥ1Ĥ2is the number of observations, we deﬁne IĤ(ā)to be the periodogram at a frequency ā∈Π2 as IĤ(ā) ≡ (2ÿ)−2Ĥ−1/barex/barex/barex/barex/barex/barexĤ1/summationdisplay.1 ĩ1=1Ĥ2/summationdisplay.1 ĩ2=1Ė(s)exp(−ăsTā)/barex/barex/barex/barex/barex/barex2 ,s≡ (ĩ1,ĩ2)T,whereă=√ −1. UsingIĤ(·)in place ofĜ(·)in (1), a standard estimator of the spectral mean parameter ĉ(ć)is then thespectral mean statistic , orspectral average , given in Riemann sum form by /hatwideĉĤ(ć) ≡ (2ÿ)2Ĥ−1/summationdisplay.1 j∈JĤć(āj,Ĥ)IĤ(āj,Ĥ), using discrete frequencies āj,Ĥ≡ (2ÿĠ1/Ĥ1,2ÿĠ2/Ĥ2)deﬁned by j≡ (Ġ1,Ġ2) ∈ JĤwith index set JĤ≡ {+−(Ĥ1−1)/2,,...,Ĥ 1−+Ĥ1/2,}×{+−(Ĥ2−1)/2,,...,Ĥ 2−+Ĥ2/2,}\{( 0,0)}denoting the (nonzero) discrete Fourier frequency grid. For calibrating tests and conﬁdence intervals, a spectral mean statistic has a large-sample normal distribution under mild conditions ( Brillinger ,2001 ,Dahlhaus ,1985 ) given by ĄĤ(ć) ≡Ĥ1/2{/hatwideĉĤ(ć)−ĉ(ć)}Ā−→N( 0,Ă2=Ă2 1+Ă2 2)asĤ→∞, (2) with Ă2 1≡Ă2 1(ć)=(2ÿ)2/uni222B.dsp Π2ć(ā)[ć(ā)+ć(−ā)]Ĝ2(ā)Ěā, Ă2 2≡Ă2 2(ć)=(2ÿ)2/uni222B.dsp Π2/uni222B.dsp Π2ć(ā1)ć(ā2)Ĝ4(ā1,ā2,−ā2)Ěā1Ěā2 where, for ā1,ā2,ā3∈Π2, Ĝ4(ā1,ā2,ā3)=(2ÿ)−3/summationdisplay.1 h1,h2,h3∈Z2cum(Ė(0),Ė(h1),Ė(h2),Ė(h3))exp/parenleftBigg −ă3/summationdisplay.1 ℓ=1hT ℓāℓ/parenrightBigg , denotes the 4th-order cumulant spectral	https://arxiv.org/abs/2504.19337v1
density. Given the complicated structure in the limit distri- bution from ( 2), one possible approach is to explore resampling for approximating the distribution ofĄĤ(ć)nonparametrically. However, estimation of the variance component Ă2 2, in particular, poses signiﬁcant challenges for bootstrap approximations to the distribution of ĄĤ(ć), as needed for the in- ference of spatial mean parameters. Specially, the spatial FDB introduced in ( Ng, Yau and Chen ,2021 ) 6 cannot capture this component in general, unless the underlying process is assumed to be Gaussian in which case this variance component Ă2 2=0 becomes zero. As a consequence, in order to improve bootstrap inference for spatial data, one must ﬁrst correctly estimate all variance components in ( 2) in the resampling mechanism used. For use in conjunction with spatial bootstrap, we next consider a spatial subsampling for the purpose of such variance estimation, as described in the following section. 3. Spatial Subsampling in Frequency Domain 3.1. Spatial subsampling variance estimators Based on the observed spatial data {Ė(s):s∈ D(Ĥ1×Ĥ2) ∩Z2}lying within the spatial sampling regionD(Ĥ1×Ĥ2) ≡ [ 1,Ĥ1] × [ 1,Ĥ2], subsampling aims to create several “smaller scale copies" of the original spatial data by using data blocks of smaller size ĘĤ≡Ĥ(Ę) 1Ĥ(Ę) 2withĤ(Ę) ġ<Ĥġ, ġ=1,2 (cf. ( Lahiri ,2003 ,Sherman and Carlstein ,1994 ,Sherman ,1996 ,1998 )) compared to Ĥ=Ĥ1Ĥ2, where the block size Ĥ(Ę) ġand the original sample size Ĥġin each direction satisfy the relation Ĥ(Ę) ġ→ ∞ andĤ−1 ġĤ(Ę) ġ→0 asĤ→ ∞ . In particular, we can deﬁne data blocks by all integer translates of a regionD(Ĥ(Ę) 1×Ĥ(Ę) 2) ≡ [1,Ĥ(Ę) 1]×[1,Ĥ(Ę) 2]that lie inside the original sampling region D(Ĥ1×Ĥ2) ≡ [1,Ĥ1]×[1,Ĥ2]; that is, each data block is of the form j+D(Ĥ(Ę) 1×Ĥ(Ę) 2)for an integer j≡ (Ġ1,Ġ2)with components Ġġ=0,...,Ĥġ−Ĥ(Ę) ġ,ġ=1,2. In total, there are, say Ĉ≡ (Ĥ1−Ĥ(Ę) 1+1)(Ĥ2−Ĥ(Ę) 2+1), such data blocks and, for simplicity, we can denote the ℓ-data block as Bℓforℓ=1,...,Ĉ . We can then deﬁne a subsample periodogram for the ℓ-th block as I(ℓ) ĩīĘ(ā) ≡ (2ÿ)−2Ę−1 Ĥ\|/summationdisplay.1 s∈Bℓ∩Z2Ė(s)ě−ăsTā\|2,ā∈Π2, ℓ=1,...,Ĉ. The corresponding subsample analogue of the mean statistic /hatwideĉĤ(ć)can then be deﬁned as /hatwideĉ(ℓ) ĩīĘ(ć) ≡ (2ÿ)2Ę−1 Ĥ/summationdisplay.1 j∈JĘć(āj,Ę)I(ℓ) ĩīĘ(āj,Ę), whereāj,ĘandJĘare deﬁned in a manner similar to āj,ĤandJĤearlier, but now based on the subsample sizes ĘĤrather than the original data size Ĥ. These lead to a collection of subsample versions Ą(ℓ) ĩīĘ(ć)of the target spectral quantity ĄĤ(ć) ≡Ĥ1/2{/hatwideĉĤ(ć)−ĉ(ć)}as Ą(ℓ) ĩīĘ(ć) ≡Ę1/2 Ĥ{/hatwideĉ(ℓ) ĩīĘ(ć)−/tildewideĉĤ(ć)}, ℓ=1,...,Ĉ, using the subsample mean /tildewideĉĤ(ć)=Ĉ−1/summationtext.1Ĉ ℓ=1/hatwideĉ(ℓ) ĩīĘ(ć)which substitutes the unknown parameter ĉ(ć). Recall that spatial variance estimation is especially important here because the variance Ă2of spec- tral mean statistics from ( 2) can be quite complex or difﬁcult to determine accurately, which is our motivation for consideration of subsampling. Consequently, we can deﬁne the subsampling variance estimator of Ă2as /hatwideĂ2 Ĥ=/hatwideĂ2 Ĥ(ć) ≡/uni222B.dsp Rİ2Ě/hatwideĂĄĤ(ć)(İ)=Ĉ−1Ĉ/summationdisplay.1 ℓ=1ĘĤ{/hatwideĉ(ℓ) ĩīĘ(ć)−/tildewideĉĤ(ć)}2, (3) Spatial Frequency Domain Bootstrap 7 which corresponds to the sample variance of the subsample statistics {Ą(ℓ) ĩīĘ(ć) ≡Ę1/2 Ĥ/hatwideĉ(ğ) ĩīĘ(ć)}Ĉ ℓ=1. Further, using subsampling, we also introduce an estimator for just the ﬁrst variance component Ă2 1(ć) in (2) as /hatwideĂ2 1,Ĥ=Ę−1 Ĥ(4ÿ2)2/summationdisplay.1 j∈JĘć(āj,Ę)(ć(āj,Ę)+ć(ā−j,Ę))1 ĈĈ/summationdisplay.1	https://arxiv.org/abs/2504.19337v1
ℓ=1(I(ℓ) ĩīĘ(āj,Ę)−/tildewideIĤ(āj,Ę))2, (4) where/tildewideIĤ(āj,Ę) ≡Ĉ−1/summationtext.1Ĉ ℓ=1I(ℓ) ĩīĘ(āj,Ę). Here in ( 4), we are isolating a component from the overall sub- sampling variance estimator Ă2 Ĥin (3) that arises due to marginal variances in subsample periodgram/braceleftbig I(ℓ) ĩīĘ(āj,Ę)/bracerightbigĈ ℓ=1. This leads to a second subsampling estimator /hatwideĂ2 2,Ĥ≡/hatwideĂ2 Ĥ−/hatwideĂ2 1,Ĥ(5) to approximate the second or remaining component Ă2 2(ć)in (2). Spatial subsampling then enables a valid estimation of both variance components Ă2 1andĂ2 2which contribute to the target distribution of ĄĤ(ć)having a limit variance of Ă2≡Ă2 1+Ă2 2. 3.2. Consistency of spatial subsampling estimators To establish the formal consistency properties for subsampling variance estimators with respect to spectral mean statistics, we use the following mild assumptions, related to the existence of limit laws. Assumption 1. {Ė(s):s∈Z2}is 4-th order stationary. Assumption 2. The limiting law in ( 2) exists for any ćof bounded variation. For two sequences j(Ĥ)≡ (Ġ(Ĥ) 1,Ġ(Ĥ) 2),k(Ĥ)≡ (ġ(Ĥ) 1,ġ(Ĥ) 2) ∈Z2of integer pairs, deﬁne Ąj(Ĥ),Ĥ(ć) andĄk(Ĥ),Ĥ(ć)as versions of ĄĤ(ć)when computed from translated spatial regions j(Ĥ)+D(Ĥ1×Ĥ2) andk(Ĥ)+D(Ĥ1×Ĥ2), instead of the region D(Ĥ1×Ĥ2) ≡ [1,Ĥ1] × [1,Ĥ2]used forĄĤ(ć)in (2). Assumption 3. For any two j(Ĥ)andk(Ĥ)integer sequences whereby \|Ġ(Ĥ) ğ−ġ(Ĥ) ğ\|/Ĥğ→∞ holds for ğ=1,2 asĤ≡Ĥ1×Ĥ2→∞ , then the following quantities are asymptotically normal and independent: /parenleftbiggĄj(Ĥ),Ĥ(ć) Ąk(Ĥ),Ĥ(ć)/parenrightbigg Ā−→N/parenleftbigg/parenleftbigg0 0/parenrightbigg ,Ă2(ć)/parenleftbigg1 0 0 1/parenrightbigg/parenrightbigg asĤ→∞ . For two integer sequences j(Ĥ),k(Ĥ)∈Z2, we use one further assumption stated below that relates to the periodogram, say Ij(Ĥ),Ĥ(ā)andIk(Ĥ),Ĥ(ā), when computed on translated sampling regions j(Ĥ)+D(Ĥ1×Ĥ2)andk(Ĥ)+D(Ĥ1×Ĥ2), respectively, for ā∈Π2≡ [−ÿ,ÿ]2. Assumption 4. LetmĤ≡ (ģĤ1,ģĤ2) ∈Z2with 0f \|ģĤğ\| fĤğ−+Ĥğ/2,,ğ=1,2 be an integer sequence such that the discrete Fourier frequencies āmĤ,Ĥconverge to some limit ā≡ (Ĉ1,Ĉ2)with 0<\|Ĉğ\|<ÿ, ğ=1,2, asĤ≡Ĥ1×Ĥ2→∞ . For any two j(Ĥ)andk(Ĥ)integer sequences whereby \|Ġ(Ĥ) ğ−ġ(Ĥ) ğ\|/Ĥğ→∞ 8 holds forğ=1,2 asĤ≡Ĥ1×Ĥ2→∞ , then the following quantities are asymptotically exponential and independent: /parenleftbiggIj(Ĥ),Ĥ(āmĤ,Ĥ) Ik(Ĥ),Ĥ(āmĤ,Ĥ)/parenrightbigg Ā−→Ĝ(ā)/parenleftbiggđ1 đ2/parenrightbigg asĤ→∞ wheređ1,đ2denote i.i.d standard exponential random variables. To comment on the conditions, Assumptions 1-2together guarantee that ĄĤ(ć)has a valid limiting law, whileĂ2(ć)is ﬁnite by Assumption 1. Assumption 3is a mild characterization of weak depen- dence in terms of the limit distributions of well-separated statistics and is also strongly connected to Assumptions 1-2. This condition states that any two copies Ąj(Ĥ),Ĥ(ć)andĄk(Ĥ),Ĥ(ć)of the statisti- cal quantity of interest ĄĤ(ć), when deﬁned on size Ĥ≡Ĥ1×Ĥ2regions that are distantly separated, should be normal and asymptotically independent as Ĥ≡Ĥ1×Ĥ2→∞ ; in particular, the spatial regions j(Ĥ)+D(Ĥ1×Ĥ2)andk(Ĥ)+D(Ĥ1×Ĥ2)for deﬁningĄj(Ĥ),Ĥ(ć)andĄk(Ĥ),Ĥ(ć)are indeed distantly separated in the sense that ğth component of the difference j(Ĥ)−k(Ĥ)diverges faster than the sample sizeĤğin each direction, ğ=1,2. Assumption 4is analogous to Assumption 3in spirit but generally weaker. Assumption 4says that spatial periodograms computed from distant regions should be asymp- totically independent and have exponential distributions up to scaling by the spectral density Ĝ; this is also a mild condition related to weak spatial dependence. Assumptions 1-4are generally intended to hold for many spatial processes, without requiring more speciﬁc characterizations of spatial depen- dence in terms of linearity or forms of mixing (cf. ( Lahiri ,2003 ,Yu, Kaiser and Nordman ,2023 )). We can next state a formal result on the consistency of subsampling variance estimators for spatial	https://arxiv.org/abs/2504.19337v1
spectral mean statistics. Theorem 3.1. Suppose Assumptions 1-4hold and the subsample size ĘĤ≡Ĥ(Ę) 1×Ĥ(Ę) 2satisﬁes 1/Ĥ(Ę) ġ+Ĥ−1 ġĤ(Ę) ġ→0 forġ=1,2 asĤ≡Ĥ1×Ĥ2→ ∞ . Then, the spatial subsampling estimators of variance components are consistent: /hatwideĂ2 1,ĤČ−→Ă2 1,/hatwideĂ2 2,ĤČ−→Ă2 2,and/hatwideĂ2 ĤČ−→Ă2≡Ă2 1+Ă2 2asĤ→∞ . The theorem above provides a device to modify and improve spatial bootstraps for approximating spectral mean statistics, particularly when such bootstraps can fail to appropriately reﬂect the spreads of such statistics. This then leads to the spatial hybrid bootstrap procedure proposed in the next section. 4. Spatial Hybrid Frequency Domain Bootstrap (HFDB) We may now describe our main resampling approach for approximating the distribution of spatial spec- tral mean statistics. Recall that, as explained in the Introduction, ( Ng, Yau and Chen ,2021 ) proposed a bootstrap for spatial data, which is generally invalid for inference about spectral averages and requires modiﬁcation for capturing the complicated variance of spectral averages. For reference, we shall term their bootstrap method as technically being a Frequency Domain Wild Bootstrap (FDWB). Recently, for the case of time series data, ( Meyer, Paparoditis and Kreiss ,2020 ) and ( Yu, Kaiser and Nordman , 2023 ) described a hybrid periodogram bootstrap (HPB) as the most advanced resampling scheme for spectral mean inference with time series in terms of broad validity and performance. Our hybrid re- sampling approach for spatial data, termed the Hybrid Frequency Domain Bootstrap (HFDB), intends types of extensions of both FDWB and HPB methods. That is, the spatial HFDB aims to combine spatial bootstrap (i.e., FDWB) with spatial subsampling (Sec. 3.1) in order to correct scaling issues Spatial Frequency Domain Bootstrap 9 in bootstrap for spatial data. This also produces a spatial analog version of HPB from time series, as explained more in the following. For setting distributional approximations for spatial spectral mean statistics, both HFDB and FDWB approaches start with a common step of mimicking the distribution of spectral statistics by a scheme of independently resampling or re-creating periodogram values based on an estimator /hatwideĜĤof the spectral densityĜ. To this end, let /hatwideĜĤrepresent a consistent estimator of the spectral density Ĝacross spatial discrete Fourier frequencies, i.e. max j∈JĤ\|/hatwideĜĤ(āj,Ĥ)−Ĝ(āj,Ĥ)\|=ĥČ(1)asĤ→∞. (6) Note that all FDB approaches involve spectral density estimation, which is true for time series (cf. ( Dahlhaus and Janas ,1996 ,Jentsch and Kreiss ,2010 ,Kreiss and Paparoditis ,2003 ,Meyer, Pa- paroditis and Kreiss ,2020 )) as well as for FDWB with spatial data as in ( Ng, Yau and Chen ,2021 ). We then deﬁne an initial bootstrap re-creation of the target spectral mean quantity ĄĤ(ć)from ( 2) as č∗ ĂĀēþ,Ĥ(ć) ≡Ĥ1/2{(2ÿ)2Ĥ−1/summationdisplay.1 j∈JĤć(āj,Ĥ)/hatwideĜĤ(āj,Ĥ)(đ∗ j−1)} (7) by drawingđ∗ jas i.i.d exponential random variables for indices j≡ (Ġ1,Ġ2) ∈ JĤinvolvingĠ1>0 or involvingĠ1=0 withĠ2>0 and then setting đ∗ −j≡đ∗ jfor the remaining indices j∈ JĤ(i.e., whereĠ1< 0 or whereĠ1=0 withĠ2<0). Note that the bootstrap approximation in ( 7) corresponds to the FDWB of (Ng, Yau and Chen ,2021 ) for spatial data and also represents a basic step in any FDB with time series (cf. ( Meyer, Paparoditis and Kreiss ,2020 )). This bootstrap	https://arxiv.org/abs/2504.19337v1
form aims to produce bootstrap versions {đ∗ j/hatwideĜĤ(āj,Ĥ)}of the periodogram ordinates {IĤ(āj,Ĥ)}(i.e., indexed over j∈ JĤ), by treating the latter as approximately independent exponential variables with respective means {Ĝ(āj,Ĥ)}. However, the problem in this bootstrap re-creation is that dependence among periodogram ordinates IĤ(āj,Ĥ), j∈ JĤ, is generally not completely ignorable so the bootstrap quantity č∗ ĂĀēþ,Ĥ(ć)cannot correctly capture the true variance Ă2≡Ă2 1+Ă2 2ofĄĤ(ć)in (2). Essentially, by independently resampling or re- constructing spatial periodogram values by {đ∗ j/hatwideĜĤ(āj,Ĥ)}, the resulting FDWB in ( 7) then estimates the second variance component Ă2 2to be zero, where the latter may not hold in general for the underlying spatial process (i.e., unless the process is assumed to be Gaussian). To overcome the above shortcoming in FDWB, a scaling adjustment with spatial subsampling can be made to deﬁne a hybrid FDB (HFDB) rendition of ĄĤ(ć)as Ą∗ ĄĂĀþ,Ĥ(ć) ≡ [Var∗{č∗ ĂĀēþ,Ĥ(ć)}+/hatwideĂ2 2,Ĥ]1/2č∗ ĂĀēþ,Ĥ(ć) [Var∗{č∗ ĂĀēþ,Ĥ(ć)}]1/2, (8) where we ﬁrst re-scale ( 7) to have unit variance at the bootstrap level using the bootstrap variance of č∗ ĂĀēþ,Ĥ(ć)}given by Var∗{č∗ ĂĀēþ,Ĥ(ć)}=Ĥ−1(4ÿ2)2/summationdisplay.1 j∈JĤć(āj,Ĥ)(ć(āj,Ĥ)+ć(ā−j,Ĥ))/hatwideĜ2 Ĥ(āj,Ĥ) and then introduce a correction Var ∗{č∗ ĂĀēþ,Ĥ(ć)}+/hatwideĂ2 2,Ĥto estimate the correct target variance Ă2≡ Ă2 1+Ă2 2; this scaling correction Var ∗{č∗ ĂĀēþ,Ĥ(ć)}+/hatwideĂ2 2,Ĥcombines the original bootstrap variance Var∗{č∗ ĂĀēþ,Ĥ(ć)}as an approximation of the ﬁrst variance component Ă2 1, with a subsampling variance estimator /hatwideĂ2 2,Ĥ=/hatwideĂ2 Ĥ−/hatwideĂ2 1,Ĥfrom ( 5) that approximates the second component Ă2 2. In this 10 manner, the hybrid bootstrap approximation Ą∗ ĄĂĀþ,Ĥ(ć)in (8) can capture the correct spread in addition to the shape of the target sampling distribution of ĄĤ(ć)for spectral inference. In the HFDB method, note that spatial subsampling plays a role in speciﬁcally estimating the variance component Ă2 2that would otherwise be missed by FDWB, while the bootstrap variance Var ∗{č∗ ĂĀēþ,Ĥ(ć)}from FDWB continues to estimate the ﬁrst component Ă2 1as implicit in ( 7). The next result provides a broad theoretical justiﬁcation for the HFDB method. Theorem 4.1. Suppose assumptions of Theorem 3.1along with ( 6). Then, for the HFDB version Ą∗ ĄĂĀþ,Ĥ(ć)ofĄĤ(ć), (a) HFDB spread is consistent for the limit variance Ă2≡Ă2 1+Ă2 2ofĄĤ(ć): Var∗{Ą∗ ĄĂĀþ,Ĥ(ć)}=Var∗{č∗ ĂĀēþ,Ĥ(ć)}+/hatwideĂ2 2,ĤČ−→Ă2asĤ→∞. (b) the HFDB approximation is consistent for the target distribution of ĄĤ(ć): sup Į∈R/barex/barex/barexČ∗(Ą∗ ĄĂĀþ,Ĥ(ć) fĮ)−Č(ĄĤ(ć) fĮ)/barex/barex/barexČ−→0 asĤ→∞, whereČ∗(·)denotes bootstrap probability induced by resampling. Because the HFDB approach combines bootstrap with spatial subsampling, Theorem 4.1shows that HFDB achieves consistency for distributional approximations of spectral means statistics under less stringent moment assumptions than previous spatial bootstraps (cf. ( Ng, Yau and Chen ,2021 )). Fur- thermore, the methodology presented above for spatial HFDB can then be applied to both Gaussian and non-Gaussian processes, while the original FDWB can only be applied to Gaussian spatial data. Numerical results in Section 5further demonstrate that the HFDB can perform well, including cases where previous bootstraps approximations (FDWB) can be seen to fail. Remark 1. As a further observation, we can decompose the target quantity ĄĤ(ć)as ĄĤ(ć) ≡Ĥ1/2{/hatwideĉĤ(ć)−ĉ(ć)}=Ĥ1/2{/hatwideĉĤ(ć)−ā(/hatwideĉĤ(ć))}+Ĥ1/2{ā(/hatwideĉĤ(ć))−ĉ(ć)}, where the ﬁrst part in the decomposition is a stochastic quantity with mean zero while the second part can be treated as a non-stochastic	https://arxiv.org/abs/2504.19337v1
bias. While the above bias decreases to zero with increasing spatial sample size Ĥ, this bias term can potentially impact approximations in small sample sizes. Therefore, in small samples, it is also possible to consider spatial subsampling for estimating this bias part as /hatwidestBias sub=Ĉ−1/summationtext.1Ĉ ℓ=1Ę1/2 Ĥ(/hatwideĉ(ℓ) sub(ć) −/hatwideĉĤ(ć)). In which case, we can use Ą∗ ĄĂĀþ,Ĥ+/hatwidestBias sub as a bootstrap approximation to ĄĤ(ć), which uses subsampling to adjust bootstrap approximations č∗ ĂĀēþ,Ĥ(ć)from ( 7) for both variance as in ( 8) as well as for bias. Remark 2. For selecting spatial block size ĘĤin practice, one may use the minimum volatility method as suggested by ( Politis, Romano and Wolf ,1999 ). The idea here is that bootstrap conﬁdence intervals, over an “appropriate" range of block sizes, should remain stable when computed as a function of block sizeĘĤ. Using this approach, one can compute bootstrap intervals for a number of block sizes and look for a region where the intervals do not change substantially according to some criterion (e.g., length). One example is to use a procedure based on minimizing a running standard deviation, as suggested by (Politis, Romano and Wolf ,1999 ). Further, one can also examine a range of block sizes with blocks D(Ĥ(Ę) 1×Ĥ(Ę) 2)deﬁned by scaling Ĥ(Ę) ġ≈ÿĤ1/4=ÿ(Ĥ1Ĥ2)1/4for someÿ>0, which can be an optimal order of spatial subsample size as suggested by ( Nordman and Lahiri ,2004 ). Spatial Frequency Domain Bootstrap 11 5. Numerical Studies of Accuracy In this section, we demonstrate the ﬁnite sample performance of the HFDB approximation of spatial spectral statistics. We compare this to the FDWB approximation of ( Ng, Yau and Chen ,2021 ) for spec- tral mean inference (cf. Section 4) and discuss the relative usefulness of the proposed HFDB method in applications. Note that our theoretical results presented in Section 3and Section 4indicate that HFDB should perform well regardless of whether the underlying process is Gaussian or not. That is, HFDB is again intended to remain valid even when the variance component Ă2 2from ( 2) is signiﬁcantly greater than zero, which represents a condition that can be encountered in practice for non-Gaussian data. In cases though where Ă2 2is exactly zero (as would occur for a Gaussian process) or approximately zero, the HFDB approach is anticipated to perform similarly to FDWB. In the following, we present numerical results for scenarios including a Gaussian process as well as a non-Gaussian process with Ă2 2differing from zero. To assess the performance of HFDB and FDWB approximations, we consider the coverage accuracy of 90% two-tailed conﬁdence intervals for the covariance parameter Ā(h)with h=(1,0)T, which are deﬁned as /parenleftBig /hatwideĀ(h)−/hatwideħ0.95Ĥ−1/2,/hatwideĀ(h)−/hatwideħ0.05Ĥ−1/2/parenrightBig , based the sample covariance estimator /hatwideĀ(h)as a case of a spectral mean statistic. The conﬁdence intervals are constructed using bootstrap quantiles /hatwideħ0.05and/hatwideħ0.95approximated from 500 Monte Carlo draws for any simulated spatial data set. To estimate the spectral density Ĝin (7) for bootstrap purposes, we used a kernel-based estimator, as detailed in ( Crujeiras and Fernandez-Casal ,2010 ). For each sample size and type of spatial process considered, we performed 1000 Monte	https://arxiv.org/abs/2504.19337v1
Carlo simulations to evaluate coverage and used 500 simulations per simulated dataset to approximate bootstrap distributions. For purposes for understanding the performance of both the FDWB and the HFDB methods in terms of coverage accuracy, below we separate numerical ﬁndings by Gaussian and non-Gaussian processes. 5.1. Results for Gaussian processes For this study, we considered weakly stationary spatial processes with a Matérn covariance function. The parametric model for the spectral density of stationary processes in the Matérn class (cf. ( Stein , 1999 )) is given by: Ĝ(ā)=č(Ă2+\|\|ā\|\|2)−ć−1, whereč>0 is a positive constant, Ăis the inverse of the range parameter, ćis the smoothness parame- ter, and\|\|·\|\|denotes theĢ2norm. For simulations, we have used ć=1, as this is often a practical choice for many climate applications. The range parameter Ăandčwere controlled to ensure that the process variance is very close to 1, thereby avoiding any nugget variance. This setup allows us to simulate real- istic spatial processes and accurately evaluate the performance of our proposed methods. We generated rectangular spatial datasets of sizes 30 ×30(Ĥ=900),50×50(Ĥ=2500), and 70×70(Ĥ=4900). Figure 1presents the coverage accuracy of 90% two-tailed conﬁdence intervals for the covariance parameter. From Figure 1, we observe that the HFDB and FDWB methods perform similarly for Gaus- sian processes ( Ă2 2=0), as perhaps expected. This similarity is anticipated to occur in this case because HFDB involves a scaling correction to FDWB that is speciﬁcally intended to be impactful when Ă2 2in (2) is non-zero. Moreover, as the overall spatial sample size Ĥincreases, the coverage accuracy im- proves across different range parameters in Figure 1, indicating that both HFDB and FDWB can be effective for Gaussian processes. 12 Figure 1 . Coverages of 90% HFDB intervals for the covariance parameter Ā(h),h=(1,0)Đ, based on either FDWB (č∗ ĂĀēþ,Ĥ(ć)) (blue line) or corrected ( Ą∗ ĄĂĀþ,Ĥ(ć)) (green line) HFDB versions with different sub- sample sizes ĘĤ, range, and sample sizes Ĥ. 5.2. Results for non-Gaussian processes In the non-Gaussian case, we ﬁrst consider a separable spatial covariance structure by deﬁning a spatial process indexed in the plane as the product of two independent processes indexed in one dimension. This simpliﬁcation facilitates analysis, providing a clearer understanding of how existing bootstrap methods extend from temporal to spatial settings, particularly with respect to higher-order cumulants and spectral density behavior. Consider two independent weakly stationary time-series {Ĕğ}and{ĕĠ} and we generate observation Ė(sğ,Ġ)at each spatial location sğ,Ġ=(ğ,Ġ)on a gridğ,Ġ=1,...√Ĥ(for√Ĥ=30,50,70) usingĖ(sğ,Ġ)=Ĕğ·ĕĠ. In this study, we have used the processes ĔĪ=0.2ĔĪ−1+ąĪ, andĕĪ=−0.7ĄĪ−1+ĄĪbased on i.i.d innovations {ąĪ}and{ĄĪ}. Figure 2presents coverage results under two distributional scenarios for these innovations. The top row of Figure 2involves innovations ąĪas Gaussian, while ĄĪfollow a standard exponential distribution; the bottom row involves both ąĪ andĄĪas standard exponential variables. Using the described model, the columns of Figure 2denote coverages over sample sizes 30 ×30(Ĥ=900),50×50(Ĥ=2500), and 70×70(Ĥ=4900). We note that in Figure 2the coverage results are based on the bias-corrected HFDB estimator, as described in Remark 1. From this ﬁgure, it is evident that the HFDB method signiﬁcantly outperforms the FDWB estimator in terms of coverage accuracy across all sample sizes,	https://arxiv.org/abs/2504.19337v1
achieving the best coverage across block sizes, and furthermore showing the best coverages as the sample size Ĥincreases. The bias correction also helps with smaller sample sizes to achieve improvement in coverages for HFDB. In contrast, the performance of the the FDWB deteriorates with increasing sample size. This issue with Spatial Frequency Domain Bootstrap 13 Figure 2 . Coverages of 90% HFDB intervals for the covariance parameter Ā(h),h=(1,0)T, based on either non- corrected (č∗ ĂĀēþ,Ĥ(ć)) (blue line) or corrected ( Ą∗ ĄĂĀþ,Ĥ(ć)) (green line) HFDB versions with different subsample sizes ĘĤ, innovations, and sample sizes Ĥ. the FDWB arises from an incorrect estimation of the limiting variance Ă2(orĂ2 2speciﬁcally) in ( 2) so that this bootstrap then leads to incorrect intervals, exhibiting extreme undercoverage in this case involving non-Gaussian processes with non-zero values of Ă2 2. We next consider another illustration of coverage accuracy using a non-Gaussian process, which serves to illustrate the behavior of spatial processes found by a nonlinear transformation to a Gaussian process. That is, we ﬁrst generated a Gaussian process using a Matérn covariance function (cf. ( Stein , 1999 )), as described earlier in Section 5.1, and applied a quartic transformation of the data to generate a non-Gaussian process which has a positive variance component Ă2 2. The quartic transformation en- sures the resulting process deviates from normality while retaining structured dependence through the underlying Matérn covariance function, allowing for a controlled study of higher-order cumulants (i.e., impactingĂ2 2) and their impact on the frequency domain bootstrap methods. Figure 3 . Coverages of 90% HFDB intervals for the covariance parameter Ā(h),h=(1,0)T, based on either non- corrected (č∗ ĂĀēþ,Ĥ(ć)) (blue line) or corrected ( Ą∗ ĄĂĀþ,Ĥ(ć)) (green line) HFDB versions with different subsample sizes ĘĤ, and sample sizes Ĥ. 14 From Figure 3, we note that the FDWB performs very poorly when the underlying data are from a non-Gaussian distribution here. In contrast, the HFDB consistently outperforms the FDWB across all sample sizes and block sizes. This superior performance of HFDB over FDWB is again due to the fact that the FDWB can fail to capture important variance components in the distribution of spectral mean statistics. By incorporating subsample variance estimation in ( 8), the HFDB achieves signiﬁcantly bet- ter results than the FDWB and achieves the desired conﬁdence level as the sample size increases. For non-Gaussian processes with Ă2 2≈0, our simulation study shows that the performance of both methods are comparable, as may expected in this case. To save space, the detailed results for these cases are presented in the Supplementary Material . 6. Application to Calibrating Spatial Isotropy Tests Here we examine the performance of bootstrap methods when applied to tests of spatial isotropy (i.e., the covariance function of the stationary spatial process is rotationally invariant). To construct a test, we follow the setup of ( Guan, Sherman and Calvin ,2004 ), which has also been adopted in ( Ng, Yau and Chen ,2021 ). In particular, we consider testing the following hypothesis: Ą0: 2Ą(hğ)=2Ą(hĠ),for all hğ,hĠ∈Λ,hğ≠hĠ,and\|\|hğ\|\|=\|\|hĠ\|\|, whereΛ≡{h1,h2,...,hģ}is a prespeciﬁed set of sites and 2	https://arxiv.org/abs/2504.19337v1
Ą(h) ≡ā(Ė(0)−Ė(h))2is the variogram at lag h; also above \|\|h\|\| ≡√ hĐh. This null hypothesis can also be written in terms of a spectral mean assessment of whether ĉ(ć)=0 holds for various spectral mean parametered deﬁned by ć(ā) ≡ {2 cos(hĐ Ġā) −2 cos(hĐ ğā)}. Lettingă≡ (2Ą(h1),...,2Ą(hģ))denote a vector of variograms in Λ, it holds that, under Ą0, there exists a full row rank matrix ýsuch thatýă=0. For example, if Λ = {(1,0),(0,1)}, thenă=(2Ą(1,0),2Ą(0,1)), and we may set ý=/bracketleftbig1−1/bracketrightbig . Exploiting this property, (Guan, Sherman and Calvin ,2004 ) proposed a test statistic ĐďĤ≡Ĥ× (ýˆăĤ)Đ(ýˆΣĎýĐ)−1(ýˆăĤ), where ˆăĤis the vector of sample variogram estimates at lag h∈Λ,Ĥis number of locations, and ˆΣĎis an estimator of the covariance matrix ΣĎof sample variograms. Under Ą0, (Guan, Sherman and Calvin , 2004 ) derived that ĐďĤĀ→Ć2 ġasĤ→∞ , whereġis the row rank of ý. Due to the slow convergence of the test statistic, ( Guan, Sherman and Calvin ,2004 ) further considered a subsampling method to determine the p-value of the test. However, since this test statistic involves spectral mean statistics in the form of variogram (or related covariance) estimators, we may study the proposed bootstrap approach in HFDB for approximating the p-values of the test. For purposes of simulating spatial data to examine size control and power in testing, we use a mean zero Gaussian process with a spherical covariance function, Ā(h) ≡  Ă2/parenleftbigg 1−3Ĩ 2č+Ĩ3 2č3/parenrightbigg +āI{h=0} if 0fĨfč, āI{h=0} otherwise,(9) whereĂ2is the partial sill parameter, čis the range parameter, āis the nugget effect, and Ĩ≡√ hĐþh is a distance related to a matrix þ, described next, as geometric anisotropy transformation. Given an anisotropy angle ăýand anisotropy ratio ăĎ, deﬁne the rotation matrix Ďand shrinking matrix Đas Ď≡/bracketleftbiggcos(ăý)sin(ăý) −sin(ăý)cos(ăý)/bracketrightbigg andĐ≡/bracketleftbigg1 0 0ăĎ/bracketrightbigg , Spatial Frequency Domain Bootstrap 15 then we may set þ≡Ď′Đ′ĐĎis a 2×2 positive deﬁnite matrix representing a geometric anisotropy transformation. A random ﬁeld with spherical covariance function as in ( 9) is called generally anisotropic unless ăĎ=1 holds, which corresponds to the isotropic case. Also, if ăý=0 holds, then the main anisotropic axes are aligned with the the standard coordinate axes (Į,į)in the plane making ăý=0 as a helpful choice in interpretation. In particular, we consider the model parameters (ā,Ă2,č,ăý,ăĎ)=(0,1,5,0,ăĎ)for different anisotropy ratio ăĎ. Also, we set Λ ={h1,h2}forh1≡ (1,0)and h2≡ (0,1)along with ă= (2Ą(h1),2Ą(h2))Đ, andý=/bracketleftbig1−1/bracketrightbig so that the test statistic form may be written as ĐďĤ≡Ĥ× [2 ˆĄ(h1) −2 ˆĄ(h2)]2(ýˆΣĎýĐ)−1, where ˆĄ(hğ),ğ=1,2, are sample variograms and where ˆΣĎmay be determined by subsampling or some other device. However, since (ýˆΣĎýĐ)−1is only a real-valued factor, we may consider a related, but simpler, test statistic ĐďĤ=Ĥ×(2 ˆĄ(1,0)−2 ˆĄ(0,1))2=Ĥ/hatwideĉ2 Ĥ(ć) to assess the null hypothesis of isotropy, which involves examining the square /hatwideĉ2 Ĥ(ć)of a spectral mean statistic /hatwideĉĤ(ć)based onć(ā) ≡ { 2 cos(hĐ 1ā) −2 cos(hĐ 2ā)}that should be near zero un- der isotropy. To consider different resampling approximations for calibrating this test statistic, we examine subsampling as in ( Guan, Sherman and Calvin ,2004 ) as well as FDWB from ( Ng, Yau and Chen ,2021 ), and the proposed HFDB	https://arxiv.org/abs/2504.19337v1
here. Both FDWB and HFDB versions of this test statis- tic are computed as [č∗ ĂĀēþ,Ĥ(ć)]2and[Ą∗ ĄĀēþ,Ĥ(ć)]2, respectively, upon applying ć(ā) ≡ {2 cos(hĐ 1ā) −2 cos(hĐ 2ā)}in (7) and ( 8); note that bootstrap statistics are then centered to have mean zero which mimics how [ĄĤ(ć) ≡√Ĥ[/hatwideĉĤ(ć) −ĉĤ(ć)]behaves under null hypothesis with ĉĤ(ć)=0. In the simulation study, we have used the spherical covariance function as in ( 9) to generate a mean- zero Gaussian spatial process, following ( Guan, Sherman and Calvin ,2004 ) and ( Ng, Yau and Chen , 2021 ). To construct a mean-zero non-Gaussian spatial process, we use the Matérn covariance with pa- rameters(Ă,ć)=(1/3,1), as described in Section 5.1. Speciﬁcally, we generate the ﬁeld by multiplying the Cholesky factor of the covariance matrix with a vector of standard exponential random variables shifted to have mean zero, thereby introducing non-Gaussianity while preserving the desired spatial dependence structure. Tables 1-2display observed rejection probabilities (based on 1000 simulations) for the Gaussian and non-Gaussian processes, respectively, based on sample sizes of 50 ×50 with a nominal testing level of 10%; similar results for a sample region 30 ×30 are presented in Supplementary Material to save space. We have used block sizes of 9 ×9 for the Gaussian process and 5 ×5 for the non- Gaussian processes to account for differences in spatial dependence and marginal behavior. The larger block size captures the smoother structure of Gaussian data, while the smaller size helps preserve the features, such as skewness and heavy tails. Table 1demonstrates that all methods perform well when the underlying distribution is Gaussian, and both FDWB and HFDB perform similarly. In particular, both maintain size close to the nominal level of 10% when the null hypothesis of isotropy is true (i.e.,ăĎ=1), and both show increasing power as ăĎdeviates from 1. However, under non-Gaussianity, FDWB fails to maintain size under isotropy to the extent that this is unsuitable for power comparison under non-Gaussian scenarios in Table 2. In contrast, both HFDB and subsampling remain effective under non-Gaussianity, with HFDB showing some advantage over subsampling as the anisotropy ratio increases. 16 Table 1. Rejection rates for Gaussian data. ăĎ Subsampling FDWB HFDB 1 0.084 0.103 0.103 1.1 0.135 0.223 0.223 1.2 0.424 0.537 0.537 1.3 0.787 0.88 0.88 1.4 0.912 0.983 0.983 1.5 0.996 1 1Table 2. Rejection rates for Non-Gaussian data.a ăĎ Subsampling HFDB 1 0.078 0.1 1.2 0.112 0.17 1.3 0.391 0.497 1.4 0.693 0.77 1.5 0.907 0.972 1.6 0.993 1 aFor non-Gaussian data, FDWB fails to maintain the size at the 10% level, with a rejection rate of 0 .243 forăĎ=1; this bootstrap becomes inappropriate for non-Gaussian data and is so excluded from Table 2. 7. Concluding Remarks The proposed spatial resampling method, called Hybrid Frequency Domain Bootstrap (HFDB), com- bines two resampling approaches in the form of subsampling and bootstrap in order to validly ap- proximation the distribution of spatial spectral mean statistics. Spectral means are fundamental spatial parameters, yet spectral statistics have complicated distributions owing to complex variances that are difﬁcult to estimate. Previous bootstrap methods, such	https://arxiv.org/abs/2504.19337v1
as the FDWB proposed by ( Ng, Yau and Chen , 2021 ), rely on speciﬁc distributional assumptions and can fail to provide valid distributional approx- imations for non-Gaussian spatial data, due to issues in correctly capturing the variances of spectral statistics for such data. In contrast, the HFDB overcomes these limitations by incorporating a scaling adjustment from spatial subsampling to correct the spread of bootstrap approximations in the frequency domain. This leads to more accurate and general estimation of the sampling distribution of spatial spec- tral mean statistics. The numerical results in Sections 5-6illustrate the ﬁnite-sample performance of the HFDB method in terms of testing and interval estimation, while the theoretical results in Sections 3-4provide some formal guarantees of consistency in HFDB distributional approximations. Importantly, our results hold under mild spatial conditions, which allows a broad scope of application for applying HFDB to spec- tral problems of assessing spatial covariance structure, with both Gaussian and non-Gaussina spatial processes. Main results along with further technical results on establishing subsampling and bootstrap for spatial data are provided in the Appendix . Appendix For notational simplicity Ęwill be used instead of ĘĤin the following proofs. All proofs are performed within the framework outlined in Section 2. Now, to prove Theorem 3.1and Theorem 4.1we will use the following lemmas which summarize all the subsampling results for spectral means that are used repeatedly in later proofs. Deﬁne ă∗(āj,Ĥ)/colonequal/hatwideĜĤ(āj,Ĥ)đ∗ j, wheređ∗ js are i.i.d standard exponential drawn using the exponential resampling mechanism. We begin by presenting the lemmas and proposi- tion necessary for proving the theorems. The proofs of the lemmas are provided last. Lemma A.1. Suppose Assumptions 1-3hold for a generic function ć:Π2→Rof bounded variation withĄĤ(ć) ≡Ĥ1/2{/hatwideĉĤ(ć) −ĉ(ć)}Ā→ N( 0,Ă2), where /hatwideĉĤ(ć),ĉ(ć), andĂ2are deﬁned analo- gously as in Section 2.2and Section 2.3. Then if subsample size Ęsatisﬁes 1 /Ĥ(Ę) ġ+Ĥ−1 ġĤ(Ę) ġ→0 for ġ=1,2 asĤ→∞ , Spatial Frequency Domain Bootstrap 17 (i) supĮ∈R/barex/barex/barexĈ−1/summationtext.1Ĉ Ģ=1I{Ę1/2(/hatwideĉ(ℓ) sub(ć)−ĉ(ć)) fĮ}−Č(ĄĤ(ć) fĮ)/barex/barex/barexČ→0; (ii) supĮ∈R/barex/barex/barexĈ−1/summationtext.1Ĉ Ģ=1I{Ę1/2(/hatwideĉ(ℓ) sub(ć)−/tildewideĉĤ(ć)) fĮ}−Č(ĄĤ(ć) fĮ)/barex/barex/barexČ→0; (iii)Ę1/2(/tildewideĉĤ(ć)−ĉ(ć))Č→0; (iv)Ĉ−1/summationtext.1Ĉ Ģ=1Ę(/hatwideĉ(ℓ) sub(ć)−ĉ(ć))2Č→Ă2(ć); (v)Ĉ−1/summationtext.1Ĉ Ģ=1Ę(/hatwideĉ(ℓ) sub(ć)−/tildewideĉĤ(ć))2Č→Ă2(ć), where /hatwideĉ(ℓ) sub(ć), and/tildewideĉĤ(ć)are analogously deﬁned as in Section 3.1. Lemma A.2. Suppose Assumptions 1-3and4hold. Then for a generic function ć:Π2→Rof bounded variation, if the subsample length Ęsatisﬁes 1 /Ĥ(Ę) ġ+Ĥ−1 ġĤ(Ę) ġ→0 forġ=1,2 asĤ→ ∞ then, 4ÿ2Ę−1/summationdisplay.1 j∈JĘć(āj,Ę)Ĉ−1Ĉ/summationdisplay.1 ℓ=1/parenleftBig I(ℓ) sub(āj,Ę)−/tildewideIĤ(āj,Ę)/parenrightBig2Č→/uni222B.dsp Π2ć(ā)Ĝ2(ā)Ěā. Proposition A.3. Forč∗ ĂĀēþ,Ĥ(ć), andĂ2 1(ć)as deﬁned in Section 4, and Section 2.3the following are true. (i) Var ∗{č∗ ĂĀēþ,Ĥ(ć)}Č→Ă2 1(ć); (ii) supĮ∈R/barex/barex/barexČ∗(č∗ ĂĀēþ,Ĥ(ć) fĮ)−¨(Į/Ă1(ć))/barex/barex/barex=ĥČ(1), whereČ∗is the probability distribution induced by resampling, Var ∗is the resampled variance, and ¨(·)is the standard normal distribution function. Proof of Proposition A.3. By the exponential resampling mechanism, we have Cov ∗(đ∗ j,đ∗ k)=I{j=−k}. Hence, with /hatwideĜĤbeing an even function it follows, Cov ∗(ă∗(āj,Ĥ),ă∗(āk,Ĥ))=/hatwideĜ2 Ĥ(āj,Ĥ)I{j=−k}and, Var∗(č∗ ĂĀēþ,Ĥ(ć))=(4ÿ2)2 Ĥ/summationdisplay.1 j,k∈JĤć(āj,Ĥ)ć(āk,Ĥ)Cov∗(ă∗(āj,Ĥ),ă∗(āk,Ĥ)) =(2ÿ)2/summationdisplay.1 j∈JĤ(2ÿ)2 Ĥć(āj,Ĥ){ć(āj,Ĥ)+ć(ā−j,Ĥ)}Ĝ2(āj,Ĥ) +(4ÿ2)2 Ĥ/summationdisplay.1 j∈JĤć(āj,Ĥ){ć(āj,Ĥ)+ć(ā−j,Ĥ)}(/hatwideĜ2 Ĥ(āj,Ĥ)−Ĝ2(āj,Ĥ)) =:ď1+ď2 Since,ď1is a Riemann sum, it immediately follows that ď1=Ă2 1(ć) +ċ(1). Using\|JĤ\|=ċ(Ĥ)and Eq. ( 6) we haveď2=ĥČ(1)which completes the proof of A.3(i). For notational ease, we deﬁne J+ Ĥ≡ {j≡ (Ġ1,Ġ2) ∈ JĤ: eitherĠ1>0 orĠ1=0 andĠ2>0}. Since {đ∗ j,j∈ J+ Ĥ}are i.i.d standard exponentially distributed and đ∗ j≡đ∗ −jforj∈ JĤwhere−j∈ J+	https://arxiv.org/abs/2504.19337v1
Ĥ, 18 thenč∗ ĂĀēþ,Ĥ(ć)can be written as č∗ ĂĀēþ,Ĥ=Ĥ−1/2/summationdisplay.1 j∈J+ĤĒ∗ j,Ĥ, (10) where, Ē∗ j,Ĥ=(2ÿ)2(ć(āj,Ĥ)/hatwideĜĤ(āj,Ĥ)+ć(−āj,Ĥ)/hatwideĜĤ(−āj,Ĥ))(đ∗ j−1),j∈ J+ Ĥ. Since,\|J+ Ĥ\|→∞ , asĤ→∞ , a conditional version of Lyapunov’s CLT can be applied to Eq. ( 10) (using the established convergence in probability of Var ∗(č∗ ĂĀēþ,Ĥ(ć))to a positive constant from (i)), i.e., we have to ﬁnd a Ą>0 such that Ĥ−(1+Ą/2)/summationdisplay.1 j∈J+Ĥā∗/parenleftBig \|Ē∗ j,Ĥ\|2+Ą/parenrightBig =ĥČ(1). (11) ChoosingĄ=1, we have /summationdisplay.1 j∈J+Ĥ\|ć(āj,Ĥ)/hatwideĜĤ(āj,Ĥ)+ć(−āj,Ĥ)/hatwideĜĤ(−āj,Ĥ)\| fĤ·sup ā∈Π\|ć(ā)\|·sup ā∈Π\|/hatwideĜĤ(ā)\|=ċČ(Ĥ), since bothćandĜ(due to absolutely summable autocovariance function) are bounded, and due to Eq. ( 6). Hence, it holds that /summationdisplay.1 j∈J+Ĥ/barex/barex/barexć(āj,Ĥ)/hatwideĜĤ(āj,Ĥ)+ć(−āj,Ĥ)/hatwideĜĤ(−āj,Ĥ)/barex/barex/barex3 =ċČ(Ĥ). (12) Since,đ∗ jare standard exponential we have ā∗(\|đ∗ j−1\|3)=12ě−1−2=ć(Say)gives us /summationdisplay.1 j∈J+ĤE∗/parenleftBig \|Ē∗ j,Ĥ\|2+Ą/parenrightBig =(4ÿ2)3ć/summationdisplay.1 j∈J+Ĥ/barex/barex/barexć(āj,Ĥ)/hatwideĜĤ(āj,Ĥ)+ć(−āj,Ĥ)/hatwideĜĤ(−āj,Ĥ)/barex/barex/barex3 =ċČ(Ĥ), due to Eq. ( 12). This shows Lyapunov condition holds for Eq. ( 11) whenĄ=1. Together with A.3(i), the assertion in A.3(ii) follows. Proof of Theorem 3.1 Using Lemma A.2, we have /hatwideĂ2 1,Ĥ(ć)Č→Ă2 1(ć). Since,ć(·)is of bounded variation, so is ć(ā)(ć(ā)+ ć(−ā)). Using Slutsky’s theorem and Lemma A.1(v) we have /hatwideĂ2 2,Ĥ(ć)=/hatwideĂ2 Ĥ(ć)−/hatwideĂ2 1,Ĥ(ć)Č→Ă2(ć)− Ă2 1(ć)=Ă2 2(ć). Then the statement of Theorem 3.1follows from (v). Proof of Theorem 4.1 (a) Now Proposition A.3(i) and Slutsky’s theorem gives us Var ∗{č∗ ĂĀēþ,Ĥ(ć)}+/hatwideĂ2 2,Ĥ(ć)Č→Ă2(ć), which completes the proof. (b) Next we prove the consistency of the distribution of the HFDB estimator. Consider an arbitrary sequence {Ĥģ}of{Ĥ}. Then it follows from Proposition A.3(i) and (ii) that there exists a further Spatial Frequency Domain Bootstrap 19 subsequence {Ĥģġ}of{Ĥģ}such that, sup Į∈R/barex/barex/barexČ∗(č∗ ĂĀēþ,Ĥ ģġ(ć) fĮ)−¨(Į/Ă1(ć))/barex/barex/barexė.ĩ.→0; Var ∗{č∗ ĂĀēþ,Ĥ ģġ(ć)}ė.ĩ.→Ă2 1(ć). It follows from Corollary 3.1(b) that Var ∗{č∗ ĂĀēþ,Ĥ ģġ(ć)}+/hatwideĂ2 2,Ĥģġ(ć)ė.ĩ→Ă2(ć)asĤģġ→∞ . Now, along the subsequence {Ĥģġ}we haveč∗ ĂĀēþ,Ĥ ģġ(ć)Ā→N( 0,Ă2 1(ć)). Thus, using Slutsky’s theo- rem gives us Ą∗ ĄĂĀþ,Ĥ ģġ(ć)Ā→N( 0,Ă2(ć)). Therefore, we have sup Į∈R/barex/barex/barexČ∗(Ą∗ ĄĂĀþ,Ĥ ģġ(ć) fĮ)−¨(Į/Ă(ć))/barex/barex/barexė.ĩ→0 asĤģġ→∞ . Since the choice of the sub-sequence is arbitrary, we have sup Į∈R/barex/barex/barexČ∗(Ą∗ ĄĂĀþ,Ĥ(ć) fĮ)−¨(Į/Ă(ć))/barex/barex/barexČ→0,asĤ→∞. Now using supĮ∈R\|Č(ĄĤ(ć) fĮ)−¨(Į/Ă(ć))\|=ĥČ(1)and triangle inequality we have, sup Į∈R/barex/barex/barexČ∗(Ą∗ ĄĂĀþ,Ĥ(ć) fĮ)−Č(ĄĤ(ć) fĮ)/barex/barex/barexČ→0 asĤ→∞ which completes the proof. Proof of Lemma A.1 To formalize our proofs, we introduce the following: for Ĩ=1,2, andĮ∈R, ĄĤ≡ĄĤ(ć);ĕ∼N( 0,Ă2(ć)),Ī(Ĩ) Ĥ≡ā(ĄĨ Ĥ),Ī(Ĩ) Ĥ,ÿ≡ā(ĄĤ−ā(ĄĤ))Ĩ, ĂĤ(Į) ≡Č(ĄĤfĮ),Ă(Į) ≡Č(ĕfĮ), /hatwideĂĤ(Į) ≡Ĉ−1/summationtext.1Ĉ ℓ=1I[Ę1/2 Ĥ(/hatwideĉ(ℓ) sub(ć) −ĉ(ć)) fĮ];/hatwideĂĤ,ÿ(Į) ≡Ĉ−1/summationtext.1Ĉ ℓ=1I[Ę1/2 Ĥ(/hatwideĉ(ℓ) sub(ć) −/tildewideĉĤ(ć)) f Į]; /hatwideĪ(Ĩ) Ĥ≡Ĉ−1/summationtext.1Ĉ ℓ=1/braceleftBig Ę1/2 Ĥ/parenleftBig /hatwideĉ(ℓ) sub(ć)−ĉ(ć)/parenrightBig/bracerightBigĨ ;/hatwideĪ(Ĩ) Ĥ,ÿ≡Ĉ−1/summationtext.1Ĉ ℓ=1/braceleftBig Ę1/2 Ĥ/parenleftBig /hatwideĉ(ℓ) sub(ć)−/tildewideĉĤ(ć)/parenrightBig/bracerightBigĨ . These deﬁnitions establish key quantities used to prove the asymptotic validity of bootstrap pro- cedures. They distinguish between population-level targets (e.g., ĉ(ć)) and sample-based estimates (e.g.,/tildewideĉĤ(ć)) to carefully track bias and variability in the resampling framework. We deﬁne Ĩ-th order central moments and empirical distributions centered around both the true and estimated targets. First we prove statement A.1(i). By notation deﬁned above, we have to show supĮ∈R\|/hatwideĂĤ(Į)−Ă(Į)\|Č→0 asĤ→ ∞ . LetC ¢Ris the set of continuity points of Ă. It sufﬁces to show /hatwideĂĤ(Į0)Č→Ă(Į0)as Ĥ→ ∞ for an arbitrary ﬁxed Į0∈ C. From this, one may take a countable, dense(in R) collection {Įğ:ğg1} ¢ C and, for any sub-sequence {Ĥģ}of{Ĥ}, extract a further sub-sequence {Ĥģġ} ≡ {Ĥġ} where, almost surely(a.s.), it holds that /hatwideĂĤģġ(Įğ) →Ă(Įğ)asĤģġ→∞ for eachğg1; the latter implies that supĮ∈R\|/hatwideĂĤģġ(Į) −Ă(Į)\|ė.ĩ.→0, which is equivalent	https://arxiv.org/abs/2504.19337v1
to supĮ∈R\|/hatwideĂĤ(Į) −Ă(Į)\|Č→0 as{Ĥģ}was arbitrary. Then statement (i) follows from a triangle inequality. Now, due to boundedness, /hatwideĂĤ(Į0)Č→Ă(Į0)is equivalent to ā(/hatwideĂĤ(Į0)−Ă(Į0))2→0, and we show the latter. 20 To control this, ﬁx ą∈ (0,1)whereąsets a threshold for spatial proximity, and for j=(Ġ1,Ġ2),k= (ġ1,ġ2)integer vectors in the plane (the ğ-th component can be 0 ,1,...,/parenleftBig Ĥğ−Ĥ(Ę) ğ/parenrightBig forğ=1,2) we split the double sum of covariances in the MSE into indices j,ksuch that: (a)\|Ġğ−ġğ\|<ąĤğfor someğ=1,2, (b) and those for which \|Ġğ−ġğ\| gąĤğfor bothğ=1,2. In case (a), the number of such dependent index pairs has an upper bound Ĥą/Ĉ. In case (b), the assumption \|Ġğ−ġğ\| Ĥ(Ę) ğ→∞ forğ=1,2, along withĤ(Ę) ğ/Ĥğ→0 (orĤğ/Ĥ(Ę) ğ→ ∞ ), implies that the covariance between I(Ąj,ĘfĮ0)and I(Ąk,ĘfĮ0)must disappear asymptotically. That is, sup \|Ġğ−ġğ\|gąĤğ ∀ğ=1,2Cov/parenleftbigI(Ąj,ĘfĮ0),I(Ąk,ĘfĮ0)/parenrightbig = sup \|Ġğ−ġğ\|gąĤğ ∀ğ=1,2/barex/barexČ(Ąj,ĘfĮ0,Ąk,ĘfĮ0)−Č(Ąj,ĘfĮ0)Č(Ąk,ĘfĮ0)/barex/barex→0, by Assumption 3. This guarantees Ā1,Ĥ(ą):= sup \|Ġğ−ġğ\|gąĤğ ∀ğ=1,2\|Č(Ąj,ĘfĮ0,Ąk,ĘfĮ0)−Ă(Į0)2\|→0, Ā2,Ĥ(ą):= sup Ġğ∈{0,1,..,(Ĥğ−Ĥ(Ę) ğ)} ∀ğ=1,2\|Č(Ąj,ĘfĮ0)/parenleftbigĂ(Į0)−Č(Ąj,ĘfĮ0)/parenrightbig\|→0. Combining, we obtain: ā/parenleftBig /hatwideĂĤ(Į0)−Ă(Į0)/parenrightBig2 fĤą Ĉ+Ā1,Ĥ(ą)+Ā2,Ĥ(ą) →ą. Sinceą>0 is arbitrary, we conclude: ā/parenleftBig /hatwideĂĤ(Į0)−Ă(Į0)/parenrightBig2 →0, completing the proof of Lemma A.1(i). Next, we prove statements (iii) - (v) together. First we would like to prove the following statements: if limĤ→∞ā(Ą2 j(Ĥ),Ĥ)=ā(ĕ2)hold for any vector of integer sequence j(Ĥ), then /hatwideĪ(1) ĤČ→Ī(1) Ĥ,/hatwideĪ(2) ĤČ→Ī(2) Ĥ,and/hatwideĪ(2) Ĥ,ÿČ→Ī(2) Ĥ,ÿasĤ→∞. (13) We show /hatwideĪ(Ĩ) ĤČ→Ī(Ĩ) Ĥ, forĨ=1,2. We would also have /hatwideĪ(2) Ĥ,ÿČ→Ī(2) Ĥ,ÿby Slutsky’s theorem. For anyÿ>0 such that ±ÿ∈ C, it follows from statement (ğ)that /hatwideĪ(Ĩ) Ĥ(ÿ) ≡1 ĈĈ/summationdisplay.1 ℓ=1ĄĨ j(ℓ),ĘI{\|Ąj(ℓ),Ę\| fÿ}=/uni222B.dsp įĨI{\|į\| fÿ}Ě/hatwideĂĤ(į)Č→ā(ĕĨI{\|ĕ\| fÿ}). (14) Spatial Frequency Domain Bootstrap 21 That is, by statement (i), for any sub-sequence {ĤĠ}of{Ĥ}, there exists again further sub-sequence {Ĥġ} ¢ {ĤĠ}where supĮ∈R\|/hatwideĂĤġ(Į) −Ă(Į)\|ė.ĩ.→0 as{Ĥġ} → 0. Then using Continuous Mapping the- orem, for a random variable ĕ∗ Ĥġ∼/hatwideĂĤġ, we have (ĕ∗ Ĥġ)ĨI{\|ĕ∗ Ĥġ\| fÿ}Ā→ĕĨI{\|ĕfÿ}, and so the cor- responding expencted values (bounded) converge as/uni222B įĨI{\|į\| fÿ}Ě/hatwideĂĤġ(į)ė.ĩ.→ā(ĕĨI{\|ĕ\| fÿ})as Ĥġ→∞ implying Eq. ( 14). For givenĄ>0 andą∈ (0,1), pick and ﬁx ÿsuch thatā(ĕĨI{\|ĕ\|>ÿ})< Ąą/3. Then it sufﬁces to show lim sup Ĥ→∞Č/parenleftBig \|/hatwideĪ(Ĩ) Ĥ−/hatwideĪ(Ĩ) Ĥ(ÿ)\|>Ą/3/parenrightBig fą, (15) from which lim sup Ĥ→∞Č/parenleftBig \|/hatwideĪ(Ĩ) Ĥ−ā(ĕĨ)\|>Ą/parenrightBig flim sup Ĥ→∞Č/parenleftBig \|/hatwideĪ(Ĩ) Ĥ(ÿ)−ā(ĕĨI{\|ĕ\| fÿ})\|>Ą/3/parenrightBig +lim sup Ĥ→∞Č/parenleftBig \|/hatwideĪ(Ĩ) Ĥ−/hatwideĪ(Ĩ) Ĥ(ÿ)\|>Ą/3/parenrightBig fą follows by Eq. ( 14)-(15) andČ(ā(ĕĨI{\|ĕ\|>ÿ})>Ą/3)=0, which gives us /hatwideĪ(Ĩ) ĤČ→ā(ĕĨ). Since we have assumed for any vector of integer sequence j(Ĥ),ā(Ą2 j(Ĥ),Ĥ) →ā(ĕ2)asĤ→∞ , we haveĪ(2) Ĥ≡ ā(Ą2 Ĥ) →ā(ĕ2)=Ă2(ć)and, by dominated convergence theorem (DCT), ā(\|ĄĤ\|) →ā(\|Ą\|)and Ī(1) Ĥ≡ā(ĄĤ) →ā(ĕ)=0 asĤ→∞ . Thus, we have established /hatwideĪ(Ĩ) ĤČ→Ī(Ĩ) ĤforĨ=1,2. By Markov’s inequality, for Ĩ=1,2, we have Č/parenleftBig \|/hatwideĪ(Ĩ) Ĥ−/hatwideĪ(Ĩ) Ĥ(ÿ)\|>Ą/3/parenrightBig fĄ 3max 1fℓfĈā/parenleftBig \|Ąj(ℓ),Ę\|ĨI{\|Ąj(ℓ),Ę\|>ÿ}/parenrightBig , so that, Eq. ( 15) follows from lim supĤ→∞max 1fℓfĈā/parenleftBig \|Ąj(ℓ),Ę\|ĨI{\|Ąj(ℓ),Ę\|>ÿ}/parenrightBig fĄą/3 which is implied by the moment conditions. Note that a contrary result lim sup Ĥ→∞max 1fℓfĈā/parenleftBig \|Ąj(ℓ),Ę\|ĨI{\|Ąj(ℓ),Ę\|>ÿ}/parenrightBig >Ąą/3 would imply existence of a sub-sequence {ĤĠ}and a positive integer ģĤĠsuch that ā/parenleftbigg \|Ąj(ģĤĠ),ĘĤĠ\|ĨI{\|Ąj(ģĤĠ),ĘĤĠ\|>ÿ}/parenrightbigg >Ąą/3 holds for each ĤĠ, which contradicts limĤĠ→∞ā/parenleftbigg \|Ąj(ģĤĠ),ĘĤĠ\|ĨI{\|Ąj(ģĤĠ),ĘĤĠ\|>ÿ}/parenrightbigg =ā(\|ĕ\|ĨI{\|ĕ\|>ÿ})<Ąą/3, that follows by DCT from Ąj(ģĤĠ),ĘĤĠĀ→ĕalong withā(\|Ąj(ģĤĠ),ĘĤĠ\|Ĩ) →ā(\|ĕ\|Ĩ)from the assumed moment conditions. 22 Now, the remaining is to show lim Ĥ→∞ā/parenleftBig Ą2 j(Ĥ),Ĥ/parenrightBig =ā(ĕ2). By 4th-order stationarity we only need to prove lim →∞ā(Ą2 Ĥ)=Ă2(ć). By Assumption 1, (Brillinger ,2001 ), and ( Fuentes ,2002 ) we have,	https://arxiv.org/abs/2504.19337v1
ā(IĤ(ā))=Ĝ(ā)+ċ(Ĥ−1),ā∈Π2(16) and forā1,ā2∈Π2, andğ=1,2, Cov(IĤ(ā1),IĤ(ā2))=  Ĝ2(ā1)+ċ(Ĥ−1) if\|Ĉ1ğ\|=\|Ĉ2ğ\|(≠0,∀ğ=1,2) (2ÿ)2 ĤĜ4(ā1,ā2,−ā2)+ċ(Ĥ−1) if\|Ĉ1ğ\|≠\|Ĉ2ğ\|.(17) The error terms are uniform in ā. Hence, forğ=1,2, Var(ĄĤ(ć))=Ĥ−1(4ÿ2)2/summationdisplay.1 j,k∈JĤć(āj,Ĥ)ć(āk,Ĥ)Cov(IĤ(āj,Ĥ),IĤ(āk,Ĥ)) =Ĥ−1(4ÿ2)2/summationdisplay.1 j∈JĤć(āj,Ĥ)/parenleftbigć(āj,Ĥ)+ć(−āj,Ĥ)/parenrightbigVar(IĤ(āj,Ĥ)) +/summationdisplay.1 j,k∈JĤ \|āĠğ\|≠\|āġğ\|ć(āj,Ĥ)ć(āk,Ĥ)Cov(IĤ(āj,Ĥ),IĤ(āk,Ĥ)) =:ď1+ď2. By Eq. ( 17) and Riemann sum form, it holds that ď1=(2ÿ)2Ĥ−14ÿ2/summationdisplay.1 j∈JĤć(āj,Ĥ)/parenleftbigć(āj,Ĥ)+ć(−āj,Ĥ)/parenrightbigĜ2(āj,Ĥ)+ċ(Ĥ−1) =(2ÿ)2/uni222B.dsp Π2ć(ā)(ć(ā)+ć(−ā))Ĝ2(ā)Ěā+ċ(Ĥ−1)=Ă2 1(ć)+ċ(Ĥ−1). Similarly, we have that ď2=Ĥ−1(4ÿ2)2/summationdisplay.1 j,k∈JĤ \|āĠğ\|≠\|āġğ\|ć(āj,Ĥ)ć(āk,Ĥ)/parenleftbigg(2ÿ)2 ĤĜ4(āj,Ĥ,āk,Ĥ,−āk,Ĥ)+ċ(Ĥ−1)/parenrightbigg =(2ÿ)2Ĥ−24ÿ2/summationdisplay.1 j,k∈JĤć(āj,Ĥ)ć(āk,Ĥ)Ĝ4(āj,Ĥ,āk,Ĥ,−āk,Ĥ) −/summationdisplay.1 j∈JĤć(āj,Ĥ)(ć(āj,Ĥ)+ć(−āj,Ĥ))Ĝ4(āj,Ĥ,āj,Ĥ,−āj,Ĥ)+ċ(Ĥ−1) =(2ÿ)2/uni222B.dsp Π2/uni222B.dsp Π2ć(ā1)ć(ā2)Ĝ4(ā1,ā2,−ā2)Ěā1Ěā2+ċ(Ĥ−1)=Ă2 2(ć)+ċ(Ĥ−1). Spatial Frequency Domain Bootstrap 23 Now, we have Var (ĄĤ(ć))=Ă2 1(ć) +Ă2 2(ć) +ċ(Ĥ−1)=Ă2(ć) +ċ(Ĥ−1). By Eq. ( 16) and bounded variation ofć(·), we also have ā(ĄĤ(ć))=Ĥ1/2/parenlefttpA/parenleftexA /parenleftbtA(2ÿ)2 Ĥ/summationdisplay.1 j∈JĤć(āj,Ĥ){Ĝ(āj,Ĥ)+ċ(Ĥ−1)}−/uni222B.dsp Π2ć(ā)Ĝ(Ĉ)ĚĈ/parenrighttpA/parenrightexA /parenrightbtA=ċ(Ĥ−1/2). Hence, we conclude that ā(ĄĤ(ć))2=Var(ĄĤ(ć))+ā2(ĄĤ(ć))=Ă2(ć)+ċ(Ĥ−1). Finally, we prove statement (ii), i.e., we will show supĮ∈R/barex/barex/barex/hatwideĂĤ,ÿ(Į)−ĂĤ(Į)/barex/barex/barexČ→0 asĤ→ ∞ . Letĕ∗ Ĥ denotes a random variable with distribution function /hatwideĂĤ, then(ĕ∗ Ĥ−/hatwideĪ(1) Ĥ)is a random variable with distribution function /hatwideĂĤ,ÿ. Because supĮ∈R/barex/barex/barex/hatwideĂĤ(Į)−Ă(Į)/barex/barex/barexČ→0 and/hatwideĪ(1) ĤČ→Ī(1) Ĥ, andĪ(1) Ĥ→ā(ĕ)= 0 guaranteed by the moment assumption, we have, for any sub-sequence {ĤĠ} ¢ {Ĥ}, there exists a further sub-sequence {Ĥġ} ¢ {ĤĠ}such thatĕ∗ ĤġĀ→ĕand/hatwideĪ(1) Ĥġ→0 hold asĤġ→ ∞ almost surely (a.s.). By Slutsky’s theorem, (ĕ∗ Ĥġ−/hatwideĪ(1) Ĥġ)Ā→ĕfollows a.s. which, because {ĤĠ}was arbitrary, implies supĮ∈R/barex/barex/barex/hatwideĂĤ,ÿ(Į)−Ă(Į)/barex/barex/barexČ→0. Statement (ii) follows from a triangle inequality. Proof of Lemma A.2 Note that, (2ÿ)2Ę−1/summationdisplay.1 j∈JĘć(āj,Ę)1 ĈĈ/summationdisplay.1 ℓ=1/parenleftBig I(ℓ) sub(āj,Ę)−/tildewideIĤ(āj,Ę)/parenrightBig2 =(2ÿ)2Ę−1/bracketleftBiggĈ/summationdisplay.1 ℓ=1ć(āj,Ę)/parenleftBigg 1 ĈĈ/summationdisplay.1 ℓ=1(I(ℓ) sub(āj,Ę))2−2Ĝ2(āj,Ę)/parenrightBigg +Ĉ/summationdisplay.1 ℓ=1ć(āj,Ę)/parenleftBig Ĝ2(āj,Ę)−(/tildewideIĤ(āj,Ę))2/parenrightBig +Ĉ/summationdisplay.1 ℓ=1ć(āj,Ę)Ĝ2(āj,Ę)/bracketrightBigg =:ď1Ĥ+ď2Ĥ+ď3Ĥ. It is immediate from the Riemann sum expression that, ď3Ĥ→/uni222B Π2ć(ā)Ĝ2(ā)asĤ→∞ . Now, pro- ceeding in the same way as ( Yu,2023 ), using Cauchy-Schwarz inequality and boundedness of ć(·)we have\|ď2Ĥ\|=ĥ(1). Also, using the moment conditions we have ď1ĤČ→0, which gives us the statement of Lemma A.2. Supplementary Material Supplementary Material for Frequency Domain Resampling for Gridded Spatial Data. In this supplementary material we present more simulation results for ﬁnite samples in support of the proposed method. 24 References BANDYOPADHYAY , S., J ENTSCH , C. and R AO, S. S. (2017). A spectral domain test for stationarity of spatio- temporal data. Journal of Time Series Analysis 38326–351. BANDYOPADHYAY , S. and L AHIRI , S. N. (2010). Asymptotic properties of Discrete Fourier transforms for spatial data. Sankhya 71221-259. BANDYOPADHYAY , S., L AHIRI , S. N. and N ORDMAN , D. J. (2015). A frequency domain empirical likelihood method for irregularly spaced spatial data. The Annals of Statistics 43519–545. BANDYOPADHYAY , S. and R AO, S. S. (2016). A test for stationarity for irregularly spaced spatial data. Journal of the Royal Statistical Society: Series B 7995-123. BERAN , J. (1992). A goodness-of-ﬁt test for time series with long range dependence. Journal of the Royal Statis- tical Society Series B: Statistical Methodology 54749–760. BRILLINGER , D. R. (2001). Time Series . Society for Industrial and Applied Mathematics, Philadelphia, PA. CRESSIE , N. A. C. (1993). Statistics for Spatial Data , revised ed. Wiley. CRUJEIRAS , R. M. and F ERNANDEZ -CASAL , R. (2010). On the estimation of the spectral density for continuous spatial processes. Statistics 44587–600. DAHLHAUS , R. (1985). Asymptotic normality of spectral estimates. Journal of Multivariate Analysis 16412–431. DAHLHAUS , R. and J ANAS , D. (1996). A frequency domain bootstrap for ratio statistics in time series	https://arxiv.org/abs/2504.19337v1
analysis. The Annals of Statistics 241934–1963. DAVISON , A. C. and H INKLEY , D. V. (1997). Bootstrap methods and their application 1. Cambridge university press. FUENTES , M. (2002). Spectral methods for nonstationary spatial processes. Biometrika 89197-210. FUENTES , M. (2006). Testing for separability of spatial-temporal covariance functions. Journal of Statistical Planning and Inference 136447-466. FUENTES , M. (2007). Approximate likelihood for large irregularly spaced spatial data. Journal of the American Statistical Association 102321-331. GUAN, Y., S HERMAN , M. and C ALVIN , J. A. (2004). A nonparametric test for spatial isotropy using subsampling. Journal of the American Statistical Association 99810–821. HALL, P., F ISHER , N. I. and H OFFMAN , B. (1994). On the nonparametric estimation of covariance functions. Annals of Statistics 222115-2134. IM, H. K., S TEIN , M. L. and Z HU, Z. (2007). Semiparametric estimation of spectral density with irregular observations. Journal of the American Statistical Association 102726–735. JENTSCH , C. and K REISS , J.-P. (2010). The multiple hybrid bootstrap—resampling multivariate linear processes. Journal of Multivariate Analysis 1012320–2345. KREISS , J.-P. and L AHIRI , S. N. (2012). Bootstrap methods for time series. In Handbook of statistics ,303–26. Elsevier. KREISS , J.-P. and P APARODITIS , E. (2003). Autoregressive-aided periodogram bootstrap for timeseries. The Annals of Statistics 311923–1955. KREISS , J.-P., P APARODITIS , E. and P OLITIS , D. N. (2011). On the range of validity of the autoregressive sieve bootstrap. Annals of Statistics 2103-2130. KREISS , J.-P. and P APARODITIS , E. (2012). The hybrid wild bootstrap for time series. Journal of the American Statistical Association 1071073–1084. KREISS , J.-P. and P APARODITIS , E. (2023). Bootstrapping Whittle estimators. Biometrika 110499–518. LAHIRI , S. N. (2003). Resampling methods for dependent data . Springer Science & Business Media. LI, W. and M CLEOD, A. (1986). Fractional time series differencing. Biometrika 73217–221. LJUNG , G. M. and B OX, G. E. (1978). On a measure of lack of ﬁt in time series models. Biometrika 65297–303. MATSUDA , Y. and Y AJIMA , Y. (2009). Fourier analysis of irregularly spaced data on Rd. Journal of the Royal Statistical Society: Series B 71191–217. MEYER , M., P APARODITIS , E. and K REISS , J.-P. (2020). Extending the validity of frequency domain bootstrap methods to general stationary processes. The Annals of Statistics 482404-2427. MILHØJ , A. (1981). A test of ﬁt in time series models. Biometrika 68177–187. NG, W. L., Y AU, C. Y. and C HEN, X. (2021). Frequency domain bootstrap methods for random ﬁelds. Electronic Journal of Statistics 156586–6632. Spatial Frequency Domain Bootstrap 25 NORDMAN , D. J. and L AHIRI , S. N. (2004). On optimal spatial subsample size for variance estimation. The Annals of Statistics 321981-2027. NORDMAN , D. J. and L AHIRI , S. N. (2006). A frequency domain empirical likelihood for short-and long-range dependence. The Annals of Statistics 3019–3050. PAPARODITIS , E. (2000). Spectral density based goodness-of-ﬁt tests for time series models. Scandinavian Jour- nal of Statistics 27143–176. PARZEN , E. (1957). On consistent estimates of the spectrum of a	https://arxiv.org/abs/2504.19337v1
stationary time series. The Annals of Mathemat- ical Statistics 329–348. POLITIS , D. N. and M CELROY , T. S. (2019). Time series: A ﬁrst course with bootstrap starter . CRC Press. POLITIS , D. N., R OMANO , J. P. and W OLF, M. (1999). Subsampling . Springer Science & Business Media. SHERMAN , M. (1996). Variance estimation for statistics computed from spatial lattice data. Journal of the Royal Statistical Society Series B: Statistical Methodology 58509–523. SHERMAN , M. (1998). Efﬁciency and robustness in subsampling for dependent data. Journal of statistical plan- ning and inference 75133–146. SHERMAN , M. and C ARLSTEIN , E. (1994). Nonparametric estimation of the moments of a general statistic com- puted from spatial data. Journal of the American Statistical Association 89496–500. STEIN , M. L. (1999). Interpolation of spatial data: some theory for kriging . Springer Science & Business Media. SUBBA RAO, S. (2018). Statistical inference for spatial statistics deﬁned in the Fourier domain. The Annals of Statistics 46469–499. TANIGUCHI , M. (1979). On estimation of parameters of Gaussian stationary processes. Journal of Applied Prob- ability 16575–591. VANHALA, M., B ANDYOPADHYAY , S., L AHIRI , S. N. and N ORDMAN , D. J. (2017). On the non-standard distribution of empirical likelihood estimators with spatial data. Journal of Statistical Planning and Inference 187109–114. VANHALA, M., B ANDYOPADHYAY , S., L AHIRI , S. N. and N ORDMAN , D. J. (2020). A general frequency domain method for assessing spatial covariance structures. Bernoulli 262463–2487. WELLER , Z. D. and H OETING , J. A. (2020). A nonparametric spectral domain test of spatial isotropy. Journal of statistical planning and inference 204177–186. YU, H. (2023). Resampling-based inference for time series in the frequency domain, PhD thesis, Iowa State University PhD Thesis. YU, H., K AISER , M. S. and N ORDMAN , D. J. (2023). A subsampling perspective for extending the validity of state-of-the-art bootstraps in the frequency domain. Biometrika asad006. Submitted to Bernoulli Supplementary Material for Frequency Domain Resampling for Gridded Spatial Data SOUVICK BERA1,a, DANIEL J. NORDMAN2,cand SOUTIR BANDYOPADHYAY1,b 1Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401, USA, aberasouvick@mines.edu ,bsbandyopadhyay@mines.edu 2Department of Statistics, Iowa State University, Ames, IA 50011, USA,cdnordman@iastate.edu Keywords: Spatial Frequency Domain Bootstrap; Periodogram; Spectral Mean 1. Numerical Studies of Accuracy For a comprehensive comparison and to complete the simulation study presented in Section 5, we further investigate the performance of the FDWB and HFDB methods for inferring spectral means for non-Gaussian spatial processes, particularly focusing on scenarios where Ă2 2≈0. This comparison aims to highlight the effectiveness of both approaches in cases where Ă2 2is very close to zero. By examining these scenarios, our overall goal is to demonstrate the robustness and accuracy of the HFDB method in a wider range of conditions. Moreover, we seek to conﬁrm that the HFDB-based estimator performs adequately in situations where Ă2 2is low, as expected theoretically. This detailed analysis further ensures that our ﬁndings are thorough and applicable to various practical situations involving non-Gaussian spatial processes. 1.1. Results for Non-Gaussian Process For	https://arxiv.org/abs/2504.19337v1
non-Gaussian processes with low Ă2 2values, we considered the Gaussian-Log-Gaussian process as described in ?. We generated datasets of sizes 30 ×30(Ĥ=900),50×50(Ĥ=2500), and 70×70(Ĥ= 4900)on regularly spaced square-shaped lattice regions. For each dataset size, we ran 500 Monte Carlo simulations for coverage and 500 simulations to generate the bootstrap distributions. This process was examined for different range parameters with the scalar parameter as 0.01 (see ?for more details on the rationale behind this choice where the scalar parameter is denoted by ć). This setup allows us to evaluate the performance of our proposed methods in a context where Ă2 2is small but not zero, providing insight into their effectiveness under these speciﬁc conditions. 1 2 Figure 1 . Coverages of 90% HFDB intervals for the covariance parameter Ā(h),h=(1,0)Đ, based on either non-corrected ( č∗ ĂĀēþ,Ĥ(ć)) (red line) or corrected ( Ą∗ ĄĂĀþ,Ĥ(ć)) (green line) HFDB versions with different subsample sizes ĘĤ, range, and sample sizes Ĥ. We note from Figure 1that both methods perform similarly when Ă2 2is very small. This contrasts with the results in Figure 2 of Section 5, where there is a signiﬁcant difference in the performance of the two methods due to a higher value of Ă2 2. We further observe that as the subsample size ĘĤand the overall sample size Ĥincrease, the coverage accuracy improves across different range parameters. 2. Application to Calibrating Spatial Isotropy Tests As described in Section 6, the simulation results for a sample region 30 ×30 , using the same parameters as before for both Gaussian and non-Gaussian processes, are based on 1000 simulations at a 10% nominal level and are as follows. HFDB: Supplementary Material 3 Table 1. Rejection rates for Gaussian data. ăĎ Subsampling FDWB HFDB 1 0.071 0.108 0.108 1.1 0.107 0.14 0.14 1.2 0.236 0.324 0.324 1.3 0.563 0.646 0.646 1.4 0.845 0.92 0.92 1.5 0.914 0.966 0.966 1.6 0.992 1 1Table 2. Rejection rates for Non-Gaussian data.a ăĎ Subsampling HFDB 1 0.066 0.105 1.2 0.101 0.127 1.4 0.325 0.464 1.6 0.62 0.743 1.8 0.873 0.922 2 0.987 1 aFor non-Gaussian data, FDB fails to maintain the size at the 10% level, with a rejection rate of 0 .277 forăĎ=1, indicating its inappropriateness for such data. The tables indicate that all frequency domain resampling methods perform well when the underly- ing distribution is Gaussian, as expected. However, under non-Gaussianity, FDWB fails to maintain size under isotropy, making it unsuitable for such scenarios. In contrast, both HFDB and subsampling remain effective even with smaller sample sizes.	https://arxiv.org/abs/2504.19337v1
SIGNAL DETECTION FROM SPIKED NOISE VIA ASYMMETRIZATION BYZHIGANG BAO1,a, KHAMANCHEONG2,cAND YUJILI1,b 1University of Hong Kong,azgbao@hku.hk;bu3011732@connect.hku.hk 2Hong Kong University of Science and Technology,ckmcheong@connect.ust.hk The signal plus noise model H=S+Yis a fundamental model in signal detection when a low rank signal Sis polluted by noise Y. In the high-dimensional setting, one often uses the leading singular values and cor- responding singular vectors of Hto conduct the statistical inference of the signalS. Especially, when Yconsists of iid random entries, the singular val- ues of Scan be estimated from those of Has long as the signal Sis strong enough. However, when the Yentries are heteroscedastic or correlated, this standard approach may fail. Especially in this work, we consider a situation that can easily arise with heteroscedastic noise but is particularly difficult to address using the singular value approach, namely, when the noise Yitself may create spiked singular values. It has been a recurring question how to distinguish the signal Sfrom the spikes in Y, as this seems impossible by examining the leading singular values of H. Inspired by the work [27], we turn to study the eigenvalues of an asymmetrized model when two samples H1=S+Y1andH2=S+Y2are available. We show that by looking into the leading eigenvalues (in magnitude) of the asymmetrized model H1H∗ 2, one can easily detect S. Unlike [27], we show that even if the spikes from Yis much larger than the strength of S, and thus the operator norm of Yis much larger than that of S, the detection is still effective. Second, we estab- lish the precise detection threshold. Third, we do not require any structural assumption on the singular vectors of S. Finally, we derive precise limiting behaviour of the leading eigenvalues of the asymmetrized model. Based on the limiting results, we propose a completely data-based approach for the detection of S. Simulation studies further show that our detection remains robust even in the heavy-tailed scenario, where only the second moments of the noise matrix entries exist. This has been widely believed to be impossible using the singular value approach. 1. Introduction. In this paper, we consider the following signal-plus-noise model H=S+Y≡S+ ΣX, where X= (xij)∈Rp×nis a random matrix with independent mean 0entries. We further denote the variance profile of the matrix by T= (tij) :=n(Var(xij)) which is the matrix with entries given by the variances of√nxij’s. Throughout the paper, we make the following assumption on the variance profile: there exist two positive constants t∗≤t∗such that t∗≤min i,jtij≤max i,jtij≤t∗. (1.1) MSC2020 subject classifications: Primary 60B20, 62G10; secondary 62H10. Keywords and phrases: signal-plus-noise model, spiked model, signal detection, asymmetrization, outlying eigenvalues. 1arXiv:2504.19450v1 [math.ST] 28 Apr 2025 2 For simplicity, we further assume that all moment of xij’s exist, although by some standard truncation technique the condition can be relaxed. Here we assume that S∈Rp×nis a fixed rank matrix and Σ∈Rp×pis a fixed rank perturbation of identity, i.e., S=kX i=1diuiv∗ i=:UDV∗, Σ =I+rX j=1σiξiθ∗ i=:I+ Ξ∆Θ∗, (1.2) where kandrare fixed nonnegative integers, and {ui},{vi},{ξi}and{θi}are 4 classes of deterministic orthonormal vectors. Here D=diag(d1,...,d k)and∆ = diag(σ1,...,σ r), where {di}′sand{σi}′sare two collections of nonnegative numbers, both	https://arxiv.org/abs/2504.19450v1
ordered in de- scending order. Throughout the paper, we always assume that Σis invertible and further assume ∥Σ−1∥op≤Kfor some constant K >0. We interpret Sas a signal, which is polluted by the noise part ΣX. Let 1be the all-one matrix. Differently from the classical setting in many previous literature where (Σ,T) = (I, 1), here we consider the general setting when the noise part ΣXitself may be heteroscedastic or even correlated. We remark here that in caseΣis diagonal, in principle we can simply write ΣXandeX, where the latter still has independent entries and a general variance profile eT, analogously to T. Even in this case, we would prefer to keep the writing ΣX, as this will give us the flexibility of choosing σi’s to be evenn-dependent, so that the eT-entries no longer satisfy the assumption (1.1). In case (Σ,T) = (I, 1), a standard approach to detect Sfrom His to investigate the leading singular values of H, as natural estimators of the counterpart of S. In the high- dimensional setting when pandnare proportional, there has been a vast of literature on the singular value approach; see [32, 11, 23, 26, 19, 47, 36, 28, 43, 37, 42] for instance. Specif- ically, as a prominent example of the famous Baik-Ben Arous-P ´ech´e (BBP) phase transition phenomenon [9], one knows that the i-th leading singular value of Hwill jump out of the sup- port of the Marchenko-Pastur law, and converges to a limiting location which is a function of di, if the diis sufficiently large. From the limiting location of the leading singular value of H, one can recover the value of di. We also refer to [10, 18, 7, 26, 50, 14, 24, 25, 12, 46, 45, 15] and the reference therein for the study of BBP transition on other models such as deformed Wigner matrices and spiked covariance matrices. The BBP transition for the signal-plus-noise model can be further extended to the case when Σis general but itself does not create any outliers in the singular value distribution of ΣX; see [33] for instance. In this case, if diis sufficiently large, one can still observe outliers in the singular value distribution of H. Nevertheless, this time, the limiting location of the leading singular value will depend on the parameters in ΣandTas well, which are often unknown and hard to estimate in applications. This prevents one from using these limiting locations to estimate di’s. As the signal plus noise model with heteroscedastic noise is ubiq- uitous (see [47, 22, 30, 31, 39, 52, 55] for instance), alternative approach to detect the signal from such general model is in high demand. In order to solve the signal detection problem for the heteroscedastic case, in [27] the authors proposed an asymmetrized non-Hermitian random matrix model for this purpose when two samples of the data are available. Actually, the work [27] primarily focuses on the deformed Wigner matrices. But the strategy can be generalized to signal-plus-noise model as well, as mentioned in [27]. More specifically, if one has two independent samples H1= S+ ΣX1andH2=S+ ΣX2, one	https://arxiv.org/abs/2504.19450v1
can turn to consider the random matrix model H1H∗ 2 or a linearization of it. Under an assumption that the operator norm of the noise part ΣXis significantly smaller than the signal part S, the authors in [27] find that the leading eigenvalue (in magnitude) of the non-Hermitian model contains the precise information of di’s, and the heteroscedasticity of ΣXonly shows up in a subleading order, and thus the unknown parameters do not matter if one aims for the first order estimate of the signals. From the SIGNAL DETECTION VIA ASYMMETRIZATION 3 mathematical point of view, the closeness between the leading eigenvalues of the signal plus non-Hermitian noise matrix model and its signal part was previously revealed in [54], where the outlier of the low rank deformation of non-Hermitian square matrix with iid entries is studied. In particular, it reveals a striking difference from the Hermitian case, where the outlier exhibits an order-1 bias toward the true signal. We also refer to [20, 16, 51, 38] for further study on the outliers of the deformed non-Hermitian random matrices. The work [27] provides a very interesting application of this closeness and also provide a non-asymptotic analysis, under general assumption on T. In this work, we will continue this line of research to show that the asymmetrization technique is powerful even various conditions in [27] are not satisfied, and it can be used to tackle other challenging questions in signal detection. Especially, we will answer a recurring question on how to detect the signal part Sfrom H when Σitself may create large spiked singular values. Although distinguishing the outlying singular values of Hcaused by the signal part Sand the noise part Σis difficult, we find again, the asymmetrization approach is effective for this purpose. Unlike the situation in [27], where detecting the existence of the signal is not an issue and only obtaining a precise estimate is challenging using the singular value approach, in our case, even detecting the existence of the signal using the singular value approach is difficult. In addition, differently from [27], we show that even when the spikes in Σis much larger than di’s inS, and thus the operator norm of the noise part is much larger than S, one can still effectively detect S. More specifically, we will provide the optimal detectability threshold,p n−1∥T∥op, fordi’s, as long as the spikes in Σis not n1/4times larger than di’s. Even when Σ =I, our threshold in terms of Tis more precise than the sufficiently large signal-to-noise ratio imposed in [27]. It particularly shows that the operator norm of the noise matrix ΣXitself is not essential in the detectability of di’s. Instead, the essential threshold is given byp n−1∥T∥opwhich could be significantly smaller than ∥ΣX∥op. We then take a step further to identify the precise fluctuation of the outliers of H1H∗ 2around the limiting location. More specifically, we will work with the following non-Hermitian random matrix Y= H1 H∗ 2 = ΣX1 (ΣX2)∗ + S S∗ =:X+S (1.3) which is a linearization of H1H∗ 2. Note that, due to the block structure,	https://arxiv.org/abs/2504.19450v1
the eigenvalues of Yare in pair. We denote the non-zero eigenvalues of Ybyλ±i≡ ±λi,i= 1,...,n ∧p. We make the convention that λ1,...,λ n∧pare those eigenvalues with arguments in (−π/2,π/2]. Further, λ1,...,λ n∧pare in descending order (in magnitude). Due to the fact that we are considering real matrix, we have another symmetry of the eigenvalues, namely, all non-real eigenvalues in {λ1,...,λ n∧p}can find their complex conjugates in this collection. Hence, regarding the ordering according to magnitude, we further make the convention that the one with argument in (0,π/2)is followed by its complex conjugate in (−π/2,0). Throughout the paper, we will be working with the following assumption. ASSUMPTION 1.1. We make the following assumptions. (i) (On dimensionality): We assume that p≡p(n)andnare comparable, i.e. there exists a constant c, such that cn≡p n→c∈(0,∞)asn→ ∞ . (ii) (On signal S): We assume that Sis a low rank matrix with rank kand admits the singular value decomposition, i.e. S=kX i=1diuiv∗ i=:UDV∗ 4 Here k≥0is fixed, D=diag(d1,...,d r)withC > d 1≥...≥dk≥0for some constant C >0, and ui’s and vi’s are the associated unit left and right singular vectors, respectively. For simplicity, we further assume that di’s are well separated, i.e., min i̸=j\|di−dj\| ≥c for some small constant c >0. (iii) (On Σ): We assume that Σis a rank rperturbation of identity, i.e. Σ =I+rX j=1σiξiθ∗ i=:I+ Ξ∆Θ∗, where r≥0is fixed; ∆ = diag(σ1,...,σ r)with∥∆∥op≤n1/4−ε0for some small but fixed ε0>0;ξi’s and θi’s are the associated unit left and right singular vectors of the low rank matrix Σ−I, respectively. In particular, ∥Σ∥opcan be much larger than the constant order. Finally, we always assume Σto be invertible and ∥Σ−1∥op≤K, for some constant K >0. (iv) (On matrix X): We assume that X= (xij)has independent entries and have general variance profile1 nT=1 n(tij). Specifically, the entries xijare real random variables with Exij= 0,Ex2 ij=tij n. We restate (1.1) below: there exist two positive constants t∗≤t∗such that t∗≤min i,jtij≤max i,jtij≤t∗. For simplicity, we further assume that all moments of xij’s exist, i.e. for any integer p≥3, there exists a constant Cp>0, such that max i,jE\|xij\|p≤Cp<∞. REMARK 1.2. Here, we remark on several possible extensions of our assumptions. First, our analysis can be extended to the case where some di’s are multiple singular values. For brevity, however, we focus on the simpler case where di’s are all distinct and well-separated. Second, another popular setting considered in the literature is (Σ,T) = ( general , 1). Here, by ”general,” we mean that Σis not necessarily a fixed-rank perturbation of I; see [49], for instance. We expect similar phenomena to occur under this setting. However, technically, it requires a separate derivation. We choose to work under Assumption 1.1 mainly to facilitate a more direct comparison with the results in [27]. Finally, it is also possible to extend our discussion to the case where the noise components of the two samples have different (Σ,T), which can occur in reality when the variance profile of the noise depends on the sampling. We leave all these extensions for future discussion. For simplicity, in the sequel, we denote σmax:=∥Σ∥op. (1.4)	https://arxiv.org/abs/2504.19450v1
Throughout this paper, we adopt the notion of stochastic domination introduced in [35], which allows a loss of boundedness, up to a small power of N, with high probability. DEFINITION 1.3. (Stochastic domination) Let X= XN(u) :N∈N,u∈U , Y = YN(u) :N∈N,u∈UN SIGNAL DETECTION VIA ASYMMETRIZATION 5 be two families of real random variables, where Yis nonnegative, and UNis a possibly N- dependent parameter set. We say that Xis stochastically dominated by Y, uniformly in u, if for arbitrary small ϵ >0, and large D >0, sup u∈UNP \|XN(u)\|> NϵYN(u) ≤N−D for large N≥N0(ϵ,D). We write X=O≺(Y)orX≺Ywhen Xis stochastically bounded byYuniformly in u. Note that in the special case when XandYare deterministic, X≺Y means for any given ϵ >0,\|XN(u)\| ≤NϵYN(u)uniformly in u, for all sufficiently large N≥N0(ϵ). In addition, we say that an event E ≡ E (N)holds with high probability if P(E)≥1−N−D for any large constant D >0when Nis large enough. Our main results are stated as follows. THEOREM 1.4 (First order limit). Suppose that Assumption 1.1 holds. If there exists a positive integer ˜k≤ksuch that d1≥...≥d˜k≥p n−1∥T∥op+δfor any small (but fixed) δ >0, then the spectrum of Yhas˜kpairs of outliers. Additionally, such outlying eigenvalues converge in probability to the strength of the signal diin magnitude. Specifically, for i= 1,...,˜k, with high probability, we have for any ϵ >0, \|λi−di\| ≤n−1 2+ϵσ2 max. REMARK 1.5. In the null case when S= 0andΣ =I, we have a random matrix Ywith independent entries and two diagonal 0 blocks. For non-Hermitian matrix with general flat variance profile, i.e., the variances of all entries are comparable to order n−1, it is known from [2] that the spectral radius of the matrix is given by the square root of the spectral radius of the variance profile. If the result extended to the case with 0 diagonal blocks, we would see that the spectral radius of Yexactly converges top n−1∥T∥op, in the null case. This indicates that our threshold is optimal. It is possible to prove the convergence of the spectral radius of Ytop n−1∥T∥opby adapting the discussions in [2, 3], for instance. But for our results on outliers, it is enough to show an upper bound of the spectral radius in the null case. In the sequel, we denote by ⃗Tk·thek-th row of T, and by ⃗T·kthek-th column of T. THEOREM 1.6 (Second order fluctuation). Suppose that Assumption 1.1 holds. Recall thatuiandviare the left and right singular vectors of signal Sassociated with singular value di. Ifdi≥p n−1∥T∥op+δ, we have the expansion to the fluctuation order λi=di+u∗ iΣX1vi+v∗ i(ΣX2)∗ui+1√ng(1 +op(1)). (1.5) Heregis independent of u∗ iΣX1vi+v∗ i(ΣX2)∗ui, and it is a centered Gaussian random variable with variance Var(g) =n d4 iX αβ" d2 i n2V(Σ∗uiu∗ iΣ,Σ∗uiu∗ iΣ)M(T,α,β ) +2 n3V(Σ∗uiu∗ iΣ,viv∗ i)α,βN(TT∗,α,β) +d2 i n2V(viv∗ i,viv∗ i)α,βM(T∗,α,β)# , 6 where M(T,α,β ) =⃗Tα· I−1 n2\|z\|2(T∗T)−1 ⃗Tβ·, N(TT∗,α,β) = TT∗ I−1 n2\|z\|2(TT∗)−1 ⃗T·β! α, V(pq∗,rs∗)α,β= (pq∗)αα(rs∗)ββ for some vector p,q,r,s ∈Rp. REMARK 1.7. Note that the fluctuation order is not necessarily order 1/√ndue to the possible n-dependence of σi’s. It could be as large as n−1/2σ2 max, depending	https://arxiv.org/abs/2504.19450v1
on Σ∗ui. The above theorem reveals a non-universal feature of the limiting distribution of the outlier, which has been previously observed in other Hermitian or non-Hermitian model; see [24, 45, 51] for instance. Particularly, the distribution of the outlier is (asymptotically) a convolution of the distribution of a linear combination of X1,X2entries and a Gaussian, which may not be Gaussian in case ui,vi,Σ∗uiare all localized, i.e., only a fixed number of components of them are nonzero. But apart from this case, the limiting distribution is still Gaussian by CLT. REMARK 1.8. Our discussion can also be easily extended to the case when there are some multiple di, i.e, some of di’s are equal. In this case, a supercritical diwith multiplicity kiwill create kicorresponding outliers λi’s. The joint distribution of these kieigenvalues is given by that of the eigenvalues of an kibykirandom matrix, whose entry distribution can also be analyzed by our derivations. As a consequence, on fluctuation level, some of these λi’s will be truly complex, i.e., the imaginary part is not 0. It can also be seen in our simulation study in Section 2. For brevity, we leave the detailed discussion to future study. 1.1. Proof Strategy. In this section, we briefly describe our proof strategy for the main results. We will start with the null case (S,Σ) = (0 ,I). We shall first show that, in this case, there is no outlier. More specifically, we denote by X0= X1 X∗ 2 , (1.6) which can be regarded as a linearization of X1X∗ 2. The variance profile of X0is V=1 n T T∗ . (1.7) This non-Hermitian matrix model can be regarded as a special case of the models consid- ered in [1], where the authors consider a general Kronecker random matrix, which is a linear combination of Kronecker products of deterministic matrices and random matrices with inde- pendent entries but general variance profile. The results in [1] shows that under rather general assumption, the spectrum of the Kronecker random matrix is contained in the self-consistent τ-pseudospectrum for any τ >0. In our case, the self-consistent τ-pseudospectrum is defined as Dτ:={z∈C:dist(0,suppρz)≤τ}, where ρzis the so called self-consistent density of states, which is the deterministic approxi- mation of the spectral distribution of X0−z’s Hermitization, i.e., Hz= X0−z X∗ 0−¯z . SIGNAL DETECTION VIA ASYMMETRIZATION 7 One calls supp ρzthe self-consistent spectrum of Hz. Hence, in order to prove that for any δ >0, the spectral radius of X0is bounded byp n−1∥T∥op+δand thus that of X1X∗ 2is bounded by n−1∥T∥op+δ, it suffices to show that z̸∈Dτfor some τ >0, given that \|z\| ≥p n−1∥T∥op+δ. Such a conclusion can be obtained by analyzing the Hermitized Dyson equation following the strategy in [2]. In our case, the Dyson equation boils down to a system of four vector equations. The analysis of the Hermitized random matrix Hzdoes not only lead to the precise upper bound of the spectral radius of X0andX1X∗ 2, it also provides the high probability upper bound of the operator norms of the Green function ∥G(λ)∥op≡ ∥(X1X∗ 2−λ)−1∥op≤C, ifλ≥n−1∥T∥op+δ. We then proceed with	https://arxiv.org/abs/2504.19450v1
studying the case S= 0 butΣis with deformation as defined in (1.2). In this case, we consider the noise part Xin (1.3), which can be regarded as a mul- tiplicative deformation of X0. Our aim is to show that as long as σmax≤n1/4−ε0for some small constant ε0>0, the results proved for X0, including the upper bound of the spectral radius and also the upper bound for Green function, still hold. That means, the multiplicative deformation does not change the spectrum of X0significantly. In particular, we show the spectral radius of Xis also bounded byp n−1∥T∥op+δw.h.p, and the operator norm of the Green function of ΣX1X∗ 2Σ∗is order 1. To this end, we shall prove that det(X −z)is uni- formly nonzero for all \|z\| ≥p n−1∥T∥op+δ. Based on the result for X0, it will be sufficient to show the smallness of the following centered quadratic form of Green function of X1X∗ 2: For any deterministic unit vectors uandv, any\|λ\| ≥n−1∥T∥op+δ, and any small constant ε >0 \|u∗G(λ)v\| ≺n−1/2+ε, G (λ) =G(λ) +1 λ. (1.8) The above type of estimates has been considered in previous literature, such as [54, 51, 20, 27], for other non-Hermitian random matrix models under various assumptions and with varying levels of precision. But all of them are carried out by rather delicate combinatorial moment method after applying a Neumann expansion. In our work, we prove (1.8) via a rather robust cumulant expansion approach, which is applied to the Green function directly. The cumulant expansion approach has been widely used to study Green function in random matrix theory; see [44, 48, 40, 34] for instance. It is a bit surprising that it has never been used to study the limiting behaviour of the outliers of non-Hermitian matrix in the literature, despite the fact that it has been widely used for the counterpart of the Hermitian matrices; see the recent survey [13] and the reference therein. After establishing these bounds for X0andX, we then turn to investigate the outliers of our model Y, created by the signal part S. The asymptotic behaviour of the outliers of Yeven- tually boils down to the analysis of the quadratic form u∗G(λ)v. From the estimate in (1.8) one can conclude Theorem 1.4 via a standard argument in [54]. Regarding the fluctuation, we find that λi−dican be written as a linear combination of Green function quadratic forms of the form u∗G(λ)vandψ∗Xaϕ,a= 1,2for various different choices of u,v,ψandϕ. A key fact is that ψ∗Xaϕis not necessarily asymptotic Gaussian, depending on the structure of uandv, butu∗G(λ)vis asymptotically Gaussian. Hence, we shall show both the asymptotic Gaussianity of the linear combinations of u∗G(λ)v’s and the asymptotic independence be- tween it and the ψ∗Xaϕterms. To this end, we turn to study the joint characteristic function ofu∗G(λ)v’s and ψ∗Xaϕ’s and study its limit. Again, the limiting characteristic function is obtained via the robust cumulant expansion approach. In contrast, even in the simpler case of iid non-Hermitian random matrix, the distribution of the outlier was previously obtained in [51, 20] via a rather involved moment method. Our approach is much more straightforward,	https://arxiv.org/abs/2504.19450v1
and robust under the general variance profile assumption, thanks to the inputs from [1]. 8 1.2. Notation. Throughout this paper, we regard nas our fundamental large parameter. Any quantities that are not explicit constant or fixed may depend on n; we often omit the argument nfrom our notation. We further use ∥A∥opfor the operator norm of a matrix A. We use CorKto denote some generic (large) positive constant. For any positive integer m, letJmKdenote the set {1,...,m }. We use 1to denote the all one vector, whose dimension is often apparent from the context, and thus is omitted from the notation. For any vector u∈Cm, we denote ⟨u⟩the average of its components. 1.3. Organization. The rest of the paper is organized as follows.In Section 2, we propose an approach to detect the signal and present some simulation study. In Section 3, we prove Theorem 1.4, based on Propositions 3.1 and 3.2. In Section 4, we prove Theorem 1.6, based on Proposition 4.1. The proofs of Propositions 3.1, 3.2 and 4.1 will be postponed to Sections 6, 7 and 5, respectively. 2. Simulation study. In this section, we present some simulation study, in order to com- pare the singular value approach based on the symmetric model and the eigenvalue approach based on the asymmetric model. We may apply our result in the following way. We define the following random domain, which is purely data based DN:=n z∈C: Rez≥λs max+N−1/2o , where we used the notation λs max:= max i\|λi\|1 arg(λi)∈[π logN,π 2] . (2.1) Our detection criterion is as follows: If λi∈DN, we identify diuiv∗ ias a signal . The rough idea is to check if a leading eigenvalue (in magnitude), λi, is significantly away from the threshold. Here we use λs max as a random approximation of the limiting spectral radius,p n−1∥T∥op, which is unknown in reality. Such an approximation is possible since the lim- iting spectral distribution of X0is rotationally symmetric, which is a consequence of the fact that our Matrix/Vector Dyson equation in (6.6) depends on \|z\|2only; see the discussion in Section 6. Hence, λs maxserves as a random approximation of the true threshold. Our choice of the additional shift N−1/2is inspired by the recent result of the iid random matrix [29]. Especially, in [29], it is known that the fluctuation of the spectral radius of iid random ma- trix (without deformations) is of order o(N−1/2). It is not clear if this fluctuation order still applies to our model with a block structure, also with a multiplicative perturbation Σ. But we expect that it should be still true at least when σmax∼1. Certainly, if diis sufficiently away from the threshold, by a constant order distance ϵ >0, as we assumed in our main the- orems, we do not have to choose a correction as delicate as N−1/2. Such a choice is mainly for the detection of those weak signals close to the threshold. The reason why we start from π/logNin the definition of λs maxis because that on fluctuation level, the outliers can indeed have nonzero imaginary part, when we	https://arxiv.org/abs/2504.19450v1
have some multiple di’s or close di’s. But in any case, the fluctuation order is no larger than N−εaccording to our assumption of σmax. Hence, the choice of the lower phase bound π/logNin (2.1) is enough to distinguish the true signals from the other eigenvalues. In the simulation study, we fix (p,n) = (800 ,2000) . We consider two settings of variance profile T, T1= 1, T 2= (Ip/2⊕1.5Ip/2) 1. We further consider following two distribution types of xij: (i)√nxij∼N(0,tij); (ii)√nxij follows Student’s t distribution with degree of freedom ν= 2.2, normalized to be mean 0 and SIGNAL DETECTION VIA ASYMMETRIZATION 9 variance tij. For the choices of S, we primarily consider the case of simple di, but we also perform simulation for multiple di. Specifically, we choose S=d1e3˜e∗ 4+d2e4˜e∗ 5+d3e5˜e∗ 6 and consider various choices of (d1,d2,d3). Here ei∈Rp,˜ej∈Rnrepresent the standard basis of respective dimensions. For Σ, we consider Σ =I+σ1e1e∗ 1+σ2e2e∗ 2. In all figures, we simply call all singular values/eigenvalues which are close to the support of the limiting singular value/eigenvalue distributions the singular values/eigenvalues , and we call those away from the support of limiting distributions the outlying singular values/ outlying eigenvalues . We determine if an eigenvalue is an outlying eigenvalue by the criteria ifλi∈DN, and we also color the negative copy of it. In all eigenvalue figures, we denote by thre sthe random threshold λs max. We determine if a singular value is an outlying one by simply checking if it is away from the theoretical right end point of (general) Marchenko Pastur law. In Fig 1, we plot the singular values of Hunder the choice (T,X) = (T1,Gaussian ), and in Fig 2, we plot the eigenvalues of Y(c.f. (1.3)) under the same choice of (T,X). In both figures, we choose simple di’s. The results for multiple di’s are presented in Fig 3 and 4. In Fig 5 and 6, we consider the setting (T,X) = (T2,Gaussian ). From these figures, we can clearly see that the eigenvalue approach can always detect the signals and also the locations of the outlying eigenvalues precisely tell the value of diwhich are above the threshold. In contrast, the spiked Σcan create additional outliers when one use the singular value approach, and thus it could be falsely detected as signals. Fig 1 Although our main theorems do not cover the heavy-tailed regime, we also present the simulation results in Fig. 7 and Fig. 8 under the choice (T,X) = (T1,Student’s t )to il- lustrate the robustness of our approach in the heavy-tailed setting. The figure for the case (T,X) = (T2,Student’s t )is similar to that of (T,X) = (T2,Gaussian ), we omit it for brevity. In this case, an additional challenge arises: even when Σ =I, there are many outlying singu- lar values due to the fatness of the distribution tail. We can also view them as spiked singular values, although they are not created by Σ. Detecting the signal from all these outliers seems impossible. In contrast, we observe that the eigenvalue approach remains robust in this	https://arxiv.org/abs/2504.19450v1
case, 10 Fig 2 Fig 3 SIGNAL DETECTION VIA ASYMMETRIZATION 11 Fig 4 Fig 5 12 Fig 6 as long as the second moments of the matrix entries exist. This distinction between the ex- treme singular values and extreme eigenvalues is supported by results from more classical random matrix models; see, for instance, [8, 53, 5, 21, 38]. 3. First order: Proof of Theorem 1.4. Recall the matrix model Yfrom (1.3). Denote the eigendecomposition of Sby S=WDW∗:=X i=±1,...,±kdiwiw∗ i, where d±i=±di, w ±i=1√ 2ui ±vi , i = 1,...,k. By considering the characteristic polynomial, if λis an eigenvalue of Ybut not an eigenvalue ofX, we can derive from det X+WDW∗−λ = det X −λ det I+DW∗(X −λ)−1W = 0 SIGNAL DETECTION VIA ASYMMETRIZATION 13 Fig 7 Fig 8 14 the following identity det I+DW∗(X −λ)−1W = 0. (3.1) We further write (X −λ)−1= λGσ(λ2) Σ X1Gσ(λ2) Gσ(λ2)X∗ 2Σ∗λGσ(λ2) , (3.2) where Gσ(z) := (Σ X1X∗ 2Σ∗−z)−1,Gσ(z) := ( X∗ 2Σ∗ΣX1−z)−1. We further set G(z) := ( X1X∗ 2−z)−1,G(z) := ( X∗ 2X1−z)−1. We denote by ρ(A)the spectral radius of a square matrix A. Recall the notion “with high probability” from Definition 1.3. We have the following proposition regarding the spectral radius and Green functions of X1X∗ 2andΣX1X∗ 2Σ∗. PROPOSITION 3.1. Under Assumption 1.1, for any small constant δ >0, we have ρ(X1X∗ 2)≤n−1∥T∥op+δ, ρ (ΣX1X∗ 2Σ∗)≤n−1∥T∥op+δ with high probability. Further, uniformly in zwith\|z\| ≥n−1∥T∥op+δ, we have for some large constant C >0, ∥G(z)∥op,∥Gσ(z)∥op,∥G(z)∥op,∥Gσ(z)∥op≤C with high probability. The proof of the above proposition will be deferred to Section 6. We then introduce the centered Green functions G(z) :=G(z) +z−1,G(z) :=G(z) +z−1. (3.3) We set for any small but fixed ε >0. qn≡qn(ε) =:n1 2−ε. (3.4) Here εshall always be chosen to be sufficiently small according to ε0in Assumption 1.1 of (iii). The following proposition provides the estimates of the quadratic forms of centered Gand G, which will be one of our main technical inputs. PROPOSITION 3.2. Under Assumption 1.1, for any \|z\| ≥n−1∥T∥op+δ, and any deter- ministic vectors u,v∈Sp−1 Candψ,ϕ∈Sn−1 C, we have ⟨u,G(z)v⟩=O≺(q−1 n),⟨ψ,G(z)ϕ⟩=O≺(q−1 n) ⟨u,X 1G(z)ϕ⟩=O≺(q−1 n),⟨ψ,G(z)X∗ 2v⟩=O≺(q−1 n). (3.5) The proof of Proposition 3.2 will be deferred to Section 7. We then proceed with the proof of Theorem 1.4. Notice that Σ−1is also a low rank perturbation of I. We then first write Gσ(z) = (Σ∗)−1 X1X∗ 2−z+z I−(Σ∗Σ)−1−1Σ−1=:G(z) +E(z). SIGNAL DETECTION VIA ASYMMETRIZATION 15 The low rank matrix I−(Σ∗Σ)−1admits a spectral decomposition I−(Σ∗Σ)−1=:LΓL∗, (3.6) where Γis a low rank diagonal matrix consisting of the non-zero eigenvalues of I−(Σ∗Σ)−1. Then, by Woodbury matrix identity, we can write X1X∗ 2−z+z I−(Σ∗Σ)−1−1= [X1X∗ 2−z+zLΓL∗]−1 =G−zGL I+zΓL∗GL−1ΓL∗G=G−zGL I−Γ +zΓL∗GL−1ΓL∗G, which gives E(z) = (Σ∗)−1−I G(z) +G(z) Σ−1−I + (Σ∗)−1−I G(z) Σ−1−I −z(Σ∗)−1GL[I−Γ +zΓL∗GL]−1ΓL∗GΣ−1. We remark here that if we replace G(z)by−1/zin the definition of E(z), it gives 0. Simi- larly, we can also write Gσ(z) =G(z)− G(z)X∗ 2LΛ(I−zL∗GLΛ)−1L∗X1G(z) =:G(z) +F(z), where Λ = ( I−Γ)−1−I. Hence, with the above notations, we can rewrite (3.2) as (X −λ)−1= λG(λ2) Σ X1G(λ2) G(λ2)X∗ 2Σ∗λG(λ2) + λE(λ2) Σ X1F(λ2) F(λ2)X∗ 2Σ∗λF(λ2). . (3.7) Recall the	https://arxiv.org/abs/2504.19450v1
definition of σmaxfrom (1.4). Further, we introduce the notation Ri(z),i= 1,2 to include all the matrices in the remainder terms after applying expansion, which satisfy that for any given unit vectors u,v ⟨u,Ri(z)v⟩=O≺(q−i nσ2i max). (3.8) By our assumption on Σ, it is easy to check σ2 max≤Cn1/2−ε0for some ε0> ε > 0. Hence, the eigenvalues of I−Γare no smaller than n−1/2+ε0. Meanwhile, the entries of L∗GLare O≺(q−1 n)according to Proposition 3.2. Consequently, ∥(I−Γ)−1ΓL∗GL∥op=o(1)with high probability, we can thus express I−Γ +zΓL∗GL−1as a Neumann series and bound the remainder terms, i.e. z(Σ∗)−1GL I−Γ +zΓL∗GL−1ΓL∗GΣ−1 =z(Σ∗)−1G(Σ∗Σ) I−(Σ∗Σ)−1 GΣ−1 −z2(Σ∗)−1G(Σ∗Σ) I−(Σ∗Σ)−1 G(Σ∗Σ) I−(Σ∗Σ)−1 GΣ−1+R2(z), (3.9) where R2(z)is defined in (3.8). With the above expansion and Proposition 3.2, we can expand G(λ2)andG(λ2)around −1/λ2in (3.7) and obtain W∗(X −λ)−1W =−1 λ+W∗ λG(λ2) Σ X1G(λ2) G(λ2)X∗ 2Σ∗λG(λ2) W +W∗ λE(λ2) 0 0 0 W+W∗R2(λ2)W, (3.10) where E(z) =A∗G+GA+A∗GA+B∗GΣ−1+ (Σ∗)−1GB+B∗GB+R(z), (3.11) 16 R(z) =−z(Σ∗)−1GBΣGBΣGΣ−1−z(Σ∗)−1GBΣGΣ−1−z(Σ∗)−1GBΣGB −zB∗GBΣGΣ−1+R2(z), Here we introduced the shorthand notations A:= Σ−1−I, B := (Σ∗Σ)(I−(Σ∗Σ)−1)Σ−1= Σ∗−Σ−1. Note that Proposition 3.2 and (3.9) imply that the first six terms in (3.11) are R1(z), while R(z)is anR2(z). Then, we have W∗h (X −λ)−1+1 λ+X λ2i W =W∗ λG(λ2) Σ X1G(λ2)) G(λ2)X∗ 2Σ∗λG(λ2) W +W∗ λE(λ2) 0 0 0 W+W∗R2(λ2)W, (3.12) With Proposition 3.2, one can follow the argument in [54] to show that λ±i(Y) =±λi(Y) =±di+O≺(q−1 nσ2 max). This concludes the proof of Theorem 1.4. 4. Second order: Proof of Theorem 1.6. In this section, we take a step further to study the fluctuation of λi. Based on (3.1) and Proposition 3.1, we further expand λiaround di, 0 = det I+DW∗(X −λ)−1W = det I+DW∗(X −di)−1W+ (di−λi)DW∗(X −di)−2W+O(\|λi−di\|2) = det I+DW∗(X −di)−1W+ (di−λi)d−2 iD+O(\|λi−di\|q−1 nσ2 max) =h 1 +diw∗ i(X −di)−1wi+ (di−λi)d−1 iiY j̸=i 1−dj di +O(\|λi−di\|q−1 nσ2 max) +O(q−2 nσ2 max∥u∗ iΣ∥2 2), (4.1) where in the third step the estimate of W∗(X − di)−2Wfollows simply from that of W∗(X −di)−1Wby applying a contour integral around di, and the last step follows from the expansion of the determinant and fact that the contribution from off-diagonal entries is small. For simplicity, we will write the error term in (4.1) as Ei:=O(\|λi−di\|q−1 nσ2 max) +O(q−2 nσ2 max∥u∗ iΣ∥2 2). In the sequel, we will also use Eito denote any generic term of the above order. From (4.1), we have λi−di=di 1 +diw∗ i(X −di)−1wi +Ei =d2 iw∗ ih (X −di)−1+1 di+X d2 ii wi−w∗ iXwi+Ei. (4.2) By further using (3.10), we have λi−di=d2 iw∗ i diG(d2 i) ΣX1G(d2 i) G(d2 i)X∗ 2Σ∗diG(d2 i) wi+d2 iw∗ i diE(d2 i) 0 0 0 wi−w∗ iXwi+Ei.(4.3) SIGNAL DETECTION VIA ASYMMETRIZATION 17 Hence, our main task is to establish the joint distribution of the following quadratic forms u∗G(z)v, ψ∗G(z)ϕ, s∗X1G(z)ζ, r∗G(z)X∗ 2η and also the linear (in X) term w∗ iXwi. We then have the following key proposition. PROPOSITION 4.1. Let\|z\| ≥n−1∥T∥op+δfor any small (but fixed) δ >0. Letmi,i= 1,...,4be fixed positive integers. For any collection of unit vectors ui,vi,ψj,ϕj,qk,γk,ηℓ,rℓ withi∈Jm1K,j∈Jm2K,k∈Jm3K,ℓ∈Jm4K, the collection of random variables n√nu∗ iG(z)vi,√nψ∗ jG(z)ϕj,√nq∗ kX1G(z)γk,√nη∗ ℓG(z)X∗ 2rℓ: i∈Jm1K,j∈Jm2K,k∈Jm3K,ℓ∈Jm4Ko (4.4) converges to the collection of jointly Gaussian variables	https://arxiv.org/abs/2504.19450v1
{Ai,Bj,Ck,Dℓ:i∈Jm1K,j∈Jm2K,k∈Jm3K,ℓ∈Jm4K} with mean 0 and the covariance structure given by Cov(Ai,Aj) =1 n\|z\|4X α,βuiαujαviβvjβ⃗T⊤ α·h I−1 n2\|z\|2T∗Ti−1⃗Tβ·, Cov(Bi,Bj) =1 n\|z\|4X α,βψiαψjαϕiβϕjβ⃗T⊤ ·αh I−1 n2\|z\|2TT∗i−1⃗T·β, Cov(Ci,Cj) =1 n2\|z\|4X α,βqiαqjαγiβγjβ TT∗h I−1 n2\|z\|2TT∗i−1⃗T·β! α, Cov(Di,Dj) =1 n2\|z\|4X α,βriαrjαηiβηjβ TT∗h I−1 n2\|z\|2TT∗i−1⃗T·β! α. (4.5) The collections {Ai},{Bj},{Ck},{Dℓ}are mutually independent. Further, the collection (4.4) is asymptotically independent of any collection of finitely many linear terms of the form√na∗Xib,i= 1,2for any deterministic unit vectors a,b. The proof of Proposition 4.1 will be stated in Section 5. Now, we proceed with the proof of Theorem 1.6. It amounts to the estimate of the variance of the Gaussian part in (4.3), as we have already shown the asymptotic independence between the Gaussian part and w∗ iXwi in Proposition 4.1. Notice that we have λi−di=d2 i(diu∗ iG(d2 i)ui+u∗ iΣX1G(d2 i)vi+v∗ iG(d2 i)X∗ 2ui+div∗ iG(d2 i)vi+diu∗ iE(d2 i)ui) −u∗ iΣX1vi−v∗ iX∗ 2Σ∗ui+Ei,(4.6) where Eis defined in (3.11). A simplification leads to u∗ iE(d2 i)ui= (Σ−1ui−ui)∗G(d2 i)ui+u∗ iG(d2 i)(Σ−1ui−ui) + (Σ−1ui−ui)∗G(d2 i)(Σ−1ui−ui) + (Σ∗ui−Σ−1ui)∗G(d2 i)Σ−1ui+ Σ−1u∗ iG(d2 i)(Σ∗ui−Σ−1ui) + (Σ∗ui−Σ−1ui)∗G(d2 i)(Σ∗ui−Σ−1ui) +Ei =u∗ iΣG(d2 i)Σ∗ui−u∗ iG(d2 i)ui+Ei.(4.7) 18 The Gaussian part of (4.2), comes from the first line of (4.6). For notational simplicity, we set M(T,α,β ) =⃗Tα· I−1 n2\|z\|2(T∗T)−1 ⃗Tβ·, N(TT∗,α,β) = TT∗ I−1 n2\|z\|2(TT∗)−1 ⃗T·β! α, V(pq∗,rs∗)α,β= (pq∗)αα(rs∗)ββ, for some vector p,q,r,s ∈Rp. Combining (4.6), (4.7), and (4.5) in Proposition 4.1, we can conclude the proof of Theorem 1.6. 5. Gaussianity of Green function quadratic forms: Proof of Proposition 4.1. In this section, we prove Proposition 4.1, based on Propositions 3.1 and 3.2. For brevity, we consider the following linear combination of the quadratic forms (5.1)Q:=√n(zu∗G(z)v+zψ∗G(z)ϕ+q∗X1G(z)γ+η∗G(z)X∗ 2r) =√n(u∗X1X∗ 2G(z)v+ψ∗X∗ 2X1G(z)ϕ+q∗X1G(z)γ+η∗G(z)X∗ 2r) In order to prove Proposition 4.1, we shall consider an arbitrary linear combination of all the quadratic forms in (4.4). Nevertheless, the following derivation for the CLT of the above Q is sufficient to show the mechanism. The generalization is straightforward. After establishing the CLT for Q, we will further comment on how to involve the linear term of Xand show the asymptotic independence simultaneously. We further define the smooth cutoff function χG≡χ(n−1TrGG∗), (5.2) where χ(t)is a smooth cutoff function which takes 1when\|t\| ≤Kfor some sufficiently large constant K > 0and takes 0when\|t\| ≥2Kand interpolates smoothly in between so that\|χ(k)(t)\| ≤Ckfor some positive constant Ckfor any fixed integer k≥0. By the upper bound of ∥G∥opin Propositions 3.1, we have \|n−1TrGG∗\| ≤Kw.h.p., when Kis large enough. Hence, χG= 1w.h.p. and χ(k) G≡χ(k)(n−1TrGG∗) = 0 w.h.p. for any fixed k≥1. Further, we have the deterministic bound ∥G∥opχ(k) G≤2CkKn, for any fixed k≥0. (5.3) Note that Q=QχGw.h.p. Hence, it suffices to establish the CLT for QχGinstead. Our aim is to establish Ef′(t) =−d2tEf(t) +o(1), where f(t) = exp(i tQχ G) (5.4) for some d>0. We will use the cumulant expansion formula [40, Lemma 1.3] to establish the above equation. The reason why we include the χGfactor is to make sure that G-factors in the derivation has a deterministic upper bound so that we can take expectations. In the sequel, for simplicity, we omit zfrom the notations of the Green functions. We further	https://arxiv.org/abs/2504.19450v1
write Ef′(t) = iEQχGf(t) = i√nE(u∗X1X∗ 2Gv+ψ∗X∗ 2X1Gϕ+q∗X1Gγ+η∗GX∗ 2r)χGf(t) =:I+II+III+IV (5.5) SIGNAL DETECTION VIA ASYMMETRIZATION 19 In the sequel, we show the estimate of the first term Iin details, and the other terms can be estimated similarly. We write via cumulant expansion [40, Lemma 1.3] I= i√nEu∗X1X∗ 2Gvf(t) = i√nX ijEx1,ijh (X∗ 2Gvu∗)jiχGf(t)i = i√nmX α=1X ijκα+1(x1,ij) α!E∂α 1,ijh (X∗ 2Gvu∗)jiχGf(t)i +R (5.6) where R=O≺(n−C)for a large constant C >0when mis large enough. The estimate can be done similarly to (7.4). As the probability for the χG̸= 1 orχ(k) G̸= 0 fork≥1is extremely small, any term involving the derivatives of χGcan be neglected easily in the cumulant expansion. Hence, in the sequel, we will focus on the derivatives of other factors, and always include the terms involving the χGderivatives into the error terms without further explanation. The first order ( α= 1) term in (5.6) reads i√nX ijE∂1 1,ijh (X∗ 2Gvu∗)jiχGf(t)i =i√nX ijtijEh −(X∗ 2G)ji(X∗ 2Gvu∗)ji+ it√n(X∗ 2Gvu∗)ji −zu∗GEijX∗ 2Gv −zψ∗GX∗ 2EijGϕ+q∗EijGγ−q∗X1GX∗ 2EijGγ−η∗GX∗ 2EijGX∗ 2r χGf(t)i =:−tX ijtijEh (X∗ 2Gvu∗)jiu∗EijX∗ 2Gv χGf(t)i +R1. (5.7) Here Eij= (δkiδℓj)k,ℓ∈Rp×n, i.e., Eij=ei˜e∗ j, where ei∈Rp,˜ej∈Rnrepresent the stan- dard basis of respective dimensions. We claim that in the RHS of the above equation, the first term is the leading term and R1is the remainder term that is negligible. More precisely, with the aid of Proposition 3.2 and Proposition 6.1, we can easily show R1=O≺(q−1 n). (5.8) We then proceed with showing that all higher order ( α≥2) terms in (5.6) are negligible. Notice that the second order term is proportional to √nX ijκ3(x1,ij)∂2 1,ijh (X∗ 2Gvu∗)jiχGf(t)i =√nX ijκ3(x1,ij)h ∂2 1,ij(X∗ 2Gvu∗)ji+ 2it∂1 1,ij(X∗ 2Gvu∗)ji∂1 1,ijQ + it(X∗ 2Gvu∗)ji∂2 1,ijQ−t2(X∗ 2Gvu∗)ji(∂1 1,ijQ)2i χGf(t) +R1 = (1) + (2) + (3) + (4) + R1, (5.9) Using (3.2) and the fact that κ3(x1,ij) =O(n−3 2), one can simply bound (1) by \|(1)\| ≤Cn−1X ij\|(X∗ 2G)ji\|2\|˜e∗ jX∗ 2Gv\|\|ui\|C.S. ≤√nq−3 n, 20 As the other terms in (5.9) all involve the derivative of Q, we first derive (5.10)∂1 1,ijQ=√nh u∗EijX∗ 2Gv−u∗X1X∗ 2GEijX∗ 2Gv +ψ∗X∗ 2EijGϕ−ψ∗X∗ 2X1GX∗ 2EijGϕ +q∗EijGγ−q∗X1GX∗ 2EijGγ−η∗GX∗ 2EijX∗ 2Gri . It is easy to obtain \|∂1 1,ijQ\|=O(√nq−1 n)by Proposition 3.2. Then for (2), by extracting an factor (X∗ 2G)jifrom ∂1 1,ij X∗ 2Gvu∗ ji, we have \|(2)\| ≤Cn−1X ij\|(X∗ 2G)ji\| X∗ 2Gvu∗ ji∂1 1,ijQ ≤Cn−1/2q−1 nX ij\|(X∗ 2G)ji\|\|(X∗ 2Gvu∗)ji\| ≤Cn−1/2q−1 np TrX∗ 2GG∗X2·v∗G∗X2X∗ 2Gv≺q−1 n, where in the last two steps we used Cauchy Schwarz inequality and the fact that the operator norms of the matrices are O≺(1). For term (3),we observe that in comparison to compare to∂1 1,ijQ,∂2 1,ijQwill bring each term an additional factor, which can be bounded crudely by O(1). Furthermore, in each term of ∂1 1,ijQ, there is a factor of the form θ∗eifor some vector θwith∥θ∥2≺1, and another factor of the form ˜e∗ jξfor some vector ξwith∥ξ∥2≺1. We further write (X∗ 2Gvu∗)ji= ˜e∗ jX∗ 2Gv·u∗ i, and apply Cauchy Schwarz inequality for the i-sum and j-sum respectively, one can easily get \|(3)\| ≤Cn−1X ij X∗ 2Gvu∗ ji∂1 1,ijQ ≤Cn−1X ij u∗ iθ∗ei˜e∗ jX∗ 2Gv˜e∗ jξ ≺n−1/2. For the term (4), since	https://arxiv.org/abs/2504.19450v1
(∂1 1,ijQ)2will produce an additional factor of n, we need to deal with this term in a finer way. Bounding one ∂1 1,ijQby√nq−1 n, and for the rest applying the same reasoning as in (3), we obtain \|(4)\| ≤Cn−1X ij X∗ 2Gvu∗ ji(∂1 1,ijQ)2 ≤Cq−1 nX ij u∗ iθ∗ei˜e∗ jX∗ 2Gv˜e∗ jξ∂1 1,ijQ ≲q−1 n. Altogether, the second order term can be bounded by q−1 n=o(1). For higher order terms, o(1)bound can be obtained similarly and actually more easily as κa+1(xij)decays by n−1/2 ifαincreases by 1. Hence, we omit the details and claim I=−tX ij\|ui\|2tijEh (X∗ 2Gvv∗G∗X2)jjχGf(t)i +O≺(q−1 n). (5.11) Similarly, for the other terms in (5.5), we also have II=−tX ij\|ϕj\|2tijEh (X2G∗ψψ∗GX∗ 2)iiχGf(t)i +O≺(q−1 n), III=−t \|z\|2X ijk\|qi\|2tijtkjEh (X1Gζζ∗G∗X∗ 1)kkχGf(t)i +O≺(q−1 n), IV=−t \|z\|2X ijk\|rj\|2tjitkiEh (X2G∗ηη∗GX∗ 2)kkχGf(t)i +O≺(q−1 n). (5.12) SIGNAL DETECTION VIA ASYMMETRIZATION 21 In the following lemma, we provide a further estimate of the leading term in (5.11). The terms in (5.12) can be estimated similarly. LEMMA 5.1. With the above notations, we have −tX ij\|ui\|2tijEh (X∗ 2Gvv∗G∗X2)jjχGf(t)i =−t n\|z\|2X α,β\|uα\|2\|vβ\|2⃗T⊤ α·h I−1 n2\|z\|2T∗Ti−1⃗Tβ·Ef(t) +O≺(q−1 n). PROOF OF LEMMA 5.1. By cumulant expansion, we have Eh (X∗ 2Gvv∗G∗X2)jjχGf(t)i =X ktkj nE∂2,kjh (Gvv∗G∗X2)kjχGf(t)i +O≺(1 nqn) =1 zX ktkj nEh (X1X∗ 2Gvv∗G∗)kk−(vv∗G∗)kk χGf(t)i +O≺(1 nqn),(5.13) where the error terms come from the estimates of the higher order terms in the expansion, and their estimate can be done similarly as before, and thus the details are omitted. Further, by applying another cumulant expansion, we have Eh (X1X∗ 2Gvv∗G∗)kkχGf(t)i =X btkb nE∂1,kbh (X∗ 2Gvv∗G∗)bkχGf(t)i =−X btkb nEh (v∗G∗X2ebe∗ bX∗ 2Geke∗ kGv) + (X∗ 2Gvv∗G∗X2)bbG∗ kk χGf(t)i +O≺(1 nqn).(5.14) Note that when we plug (5.14) to (5.13), the contribution from the first term in (5.14) is negligible, since 1 n2X k,btkjtkbv∗G∗X2ebe∗ bX∗ 2Geke∗ kGv=1 n2X ktkjv∗G∗X2Tk·X∗ 2Geke∗ kGv =X ktkjv∗Akeke∗ kGv=1 n2v∗Bjv=O≺(1 n2) where Ti·=diag({tij}n j=1),T·j=diag({tij}p i=1), and Ai,Biare some matrices that de- pends on Ti·orT·iwithO≺(1)operator norm bounds. Hence, approximating G∗ kkin (5.14) and(vv∗G∗)kkin (5.13) by −1/¯zand−\|vk\|2/¯zrespectively, we can derive from (5.13) that Eh (X∗ 2Gvv∗G∗X2)jjχGf(t)i =1 n2\|z\|2X k,btkjtkbEh (X∗ 2Gvv∗G∗X2)bbχGf(t)i +1 n\|z\|2X ktkj\|vk\|2E[f(t)] +O≺(1 nqn). A further estimate of the variances of (X∗ 2Gvv∗G∗X2)jj’s via cumulant expansion leads to Eh (X∗ 2Gvv∗G∗X2)jjχGf(t)i =Eh (X∗ 2Gvv∗G∗X2)jjχGi E[f(t)] +O≺(1 nqn) 22 =1 n2\|z\|2X k,btkjtkbEh (X∗ 2Gvv∗G∗X2)bbχGi E[f(t)] +1 n\|z\|2X ktkj\|vk\|2E[f(t)] +O≺(1 nqn). (5.15) It is also easy to see that the second equation still holds if we cancel out the E[f(t)]factors. In light of this fact, we denote by ⃗Tk·thek-th row of T, and by ⃗T·kthek-th column of T. Further, we set ⃗M= (Mj)n j=1withMj=E(nX∗ 2Gvv∗G∗X2)jj. Then we have the self consistent equation ⃗M=1 \|z\|2X k\|vk\|2 I−1 n2\|z\|2(T∗T)−1 ⃗Tk·+⃗ ε, (5.16) where ⃗ εis an error vector with ∥⃗ ε∥∞=O≺(q−1 n). Plugging (5.16) back to (5.15), we get −tX ij\|ui\|2tijEh (X∗ 2Gvv∗G∗X2)jjχGf(t)i =−t n\|z\|2X ik\|ui\|2\|vk\|2⃗T⊤ i· I−1 n2\|z\|2(T∗T)−1 ⃗Tk·Ef(t) +O≺(q−1 n). This concludes the proof of Lemma 5.1. We then proceed with the proof of Proposition 4.1 for the quadratic forms in (5.1). Simi- larly to the proof of Lemma 5.1, we can also estimate other terms in (5.12). In summary, we have I=−t	https://arxiv.org/abs/2504.19450v1
n\|z\|2X ik\|ui\|2\|vk\|2⃗T⊤ i· I−1 n2\|z\|2(T∗T)−1 ⃗Tk·Ef(t) +O≺(q−1 n), II=−t n\|z\|2X jk\|ϕj\|2\|ψk\|2⃗T⊤ ·j I−1 n2\|z\|2(TT∗)−1 ⃗T·kEf(t) +O≺(q−1 n), III=−t n2\|z\|4X ik\|qi\|2\|ζk\|2 TT∗h I−1 n2\|z\|2(TT∗)i−1⃗T·k iEf(t) +O≺(q−1 n), IV=−t n2\|z\|4X jk\|rj\|2\|ηk\|2 TT∗h I−1 n2\|z\|2(TT∗)i−1⃗T·k jEf(t) +O≺(q−1 n). Plugging the above estimates into (5.5), we get an estimate of the form in (5.4), and thus a CLT follows for the particular linear combination in (5.1). In order to prove the general joint CLT for the quadratic forms in (4.4), we shall consider the characteristic function of more general linear combination of the form √n zm1X i=1c1iu∗ iG(z)vi+zm2X j=1c2jψ∗ jG(z)ϕj+m3X k=1c3kq∗ kX1G(z)γk+m4X ℓ=1c4ℓη∗ ℓG(z)X∗ 2rℓ . The derivation is a straightforward extension of that is done for Qin (5.1). For brevity, we omit the details and claim (4.5). What remains is to show the asymptotic independence of the Green function quadratic forms in (4.4) and the linear (in Xi) term a∗Xib. It can be proved via a slight modification SIGNAL DETECTION VIA ASYMMETRIZATION 23 of the above derivation of the CLT. We illustrate the necessary modification as follows, again based on the simple linear combination Qdefined in (5.1). We now involve the term of the form a∗X1b, for instance, and define f(t,s) = exp(i tQχ G+√nisa∗X1b). We can add the term of the form c∗X2das well. But for brevity, we restrict ourselves to the above quantity. Instead of (5.4), we now need to show ∂ ∂tEf(t,s) =−d2tEf(t,s) +o(1). (5.17) Solving the above equation with Ef(0,s) =Eexp(√nisa∗X1b)gives Ef(t,s) = exp( −d2t2 2)Eexp(√nisa∗X1b) +o(1) which proves the asymptotic Gaussianity of Q, and the asymptotic independence between Q and√na∗X1bsimultaneously. Hence, it suffices to illustrate how to adapt the proof of (5.17) from that of (5.4). Similarly to (5.5), we can write ∂ ∂tEf(t,s) = iEQf(t,s) = i√nE(u∗X1X∗ 2Gv+ψ∗X∗ 2X1Gϕ+q∗X1Gγ+η∗GX∗ 2r)χGf(t,s) =:eI+fII+gIII+fIV . It suffices to illustrate the estimate of eI, as an adapt of the estimate of Iin (5.5). The other terms can be estimated similarly. We again apply cumulant expansion as (5.6). In the first order term of the cumulant expansion, we will have all terms analogous to those in (5.7), with an additional term which involves the derivative of the newly added√na∗X1b. This terms reads iX ijEh (X∗ 2Gvu∗)jiaibjχGf(t,s)i = iE[u∗a·b∗X∗ 2Gvχ G·f(t,s)] =O≺(q−1 n), where we used Proposition 3.2. The derivative of the√na∗X1bterm will also show up in the higher order terms in the cumulant expansion, but the related terms can all be estimated easily with the aid of Proposition 3.2 and Cauchy Schwarz inequality. Hence, we omit the details and conclude (5.17). 6. Spectral norm estimate: Proof of Proposition 3.1. In this section we will prove Proposition 3.1. We will first prove the result on X1X∗ 2without using Proposition 3.2; see Proposition 6.1 and its proof below. Then, we prove those bounds for ΣX1X∗ 2Σ∗with the aid of Proposition 3.2; see Proposition 6.3 and its proof below. We remark here that the proof of Proposition 3.2 in Section 7 will need those bounds in Proposition 6.1. Recall the linearization of X1X∗ 2from (1.6) and its variance profile (1.7). For simplicity, in this section we denote	https://arxiv.org/abs/2504.19450v1
by T=n−1T (6.1) the variance profile of X1andX2and we recall the flatness assumption in (1.1). We will follow the strategy developed in [1, 2, 3]. Especially, our matrix X0can be re- garded as a special case of the general model considered in [1]. Based on the result for X0, we then derive the results for the model Xvia a perturbation argument. 24 A standard strategy to study the spectrum of a non-Hermitian random matrix is via the Girko’s Hermitization/linearization. Specifically, the spectrum of the (n+p)×(n+p)-matrix X0can be studied by analyzing the following 2(n+p)×2(n+p)Hermitian matrix Hz= X0−z X∗ 0−¯z , where zis a generic complex number. It is known that the possible eigenvalues of X0around zcan be studied via the spectrum of Hzaround 0. Heuristically, if the eigenvalues of Hzare aways from 0, the eigenvalues of X0will be away from z. The spectrum of Hzcan then be studied via its Green function Gz(ω) = (Hz−ω)−1. Following the study in [1, 2, 3], when the variance profile Vis general, one needs to consider the solution to the following Matrix Dyson Equation, which shall be regarded as an approximation of the Green function Gz(iη) for any η >0. It is defined as (6.2) −Mz(iη)−1=iη1−Az+V[Mz(iη)] where in our case Az= 0−z −¯z0 . Here the functional V[·]is defined as V[W] = diag(Vw2) 0 0 diag(V∗w1) for any 2(n+p)×2(n+p)matrix Wsuch that W= (wij)2(n+p) i,j=1∈C2(n+p)×2(n+p), w1= (wii)n+p i=1∈Cn+p, w 2= (wii)2(n+p) i=n+p+1∈Cn+p. It is shown in [41] that (6.2) has a unique solution under the constraint that ImMz:= (Mz− (Mz)∗)/2iis positive definite. According to [4], we define the self-consistent density of states ρzofHzas the unique measure whose Stietjes transform is1 2(p+n)TrMz(ω). More precisely, the solution Mz(iη)to (6.2) has the Stieltjes transform representation Mz(iη) =Z RV(dx) x−iη whereVis a matrix-valued, compactly supported measure on R. Then, ρz(dx) :=1 2(n+p)TrV(dx). Letmz j(iη)be the j-th diagonal entry of the matrix Mz(iη). For any τ >0, define the set Dτ:={z∈C:dist(0,suppρz)≤τ} and (6.3) eDτ:={z: limsup η→01 ηmax j\|Immz j(iη)\| ≥1 τ} Dτis called the self-consistent τ-pseudospectrum of X0. It is shown in [1] that eigenvalues ofX0will concentrate on Dτfor any fixed τ >0. It is also shown in [1] that DτandeDτare comparable in the sense that for any τ >0, we have Dτ1⊂eDτ⊂Dτ2for certain τ1,τ2>0. SIGNAL DETECTION VIA ASYMMETRIZATION 25 Further, in our case, the matrix equation (6.2) implies that Mzhas the block structure such that its top-left, top-right, lower-left and lower-right (n+p)×(n+p)blocks are all diagonal matrices. After simplification, the Matrix Dyson Equation (6.2) admit the following form for some vectors uandv, (6.4) Mz(iη) =diag(iu)−zdiag(u η+V∗u) −¯zdiag(v η+Vv) diag(iv) Therefore, to determine eDτ, as well as Dτ, we only need to analyze the following coupled vector equations derived from (6.2) and (6.4) via Schur complement 1 u=η+Vv+\|z\|2 η+V∗u, 1 v=η+V∗u+\|z\|2 η+Vv. (6.5) Here u,v∈Rp+n +are the unique solutions to the vector equations. The uniqueness and exis- tence of the positive solutions to this vector equation is a consequence of the solution to the Matrix Dyson Equation. Bijection between the solution to the Matrix Dyson Equation (6.2) with positive definite ImMzand the positive solutions	https://arxiv.org/abs/2504.19450v1
to the vector equation (6.5) is proved in [2]. One can check from (6.4) that the diagonal part of Mzis in fact purely imaginary and the imaginary part Immz j(iη)form the vector uandv. Using the block structure of V, one can further write the system of vector equations (6.5) as a system of four equations, with the notation (6.1), 1 u1=η+Tv2+\|z\|2 η+Tu2 1 u2=η+T∗v1+\|z\|2 η+T∗u1 1 v1=η+Tu2+\|z\|2 η+Tv2 1 v2=η+T∗u1+\|z\|2 η+T∗v1(6.6) Here u=u1 u2 , v =v1 v2 , u 1,v1∈Rp +, u 2,v2∈Rn +. According to Theorem 2.4 and Remark 2.5 (iv) in [1] , to get the upper bound ρ(X0)≤p ρ(V) +δ, (6.7) it suffices to show that limsup η→01 ηmax j\|Immz j(iη)\|<1 δ′(6.8) for some δ′>0, given that \|z\| ≥p ρ(V) +δ. If (6.8) holds, by the equivalence of Dτand eDτ, the eigenvalues of Hzis away from zero by a distance of δ′. We can then conclude from Theorem 4.7 in [1] that Hzis invertible and the resolvent at 0 is bounded by a constant, if (6.8) is granted. Specifically, we will have the following bound on the operator norm of (X0−z)−1and the Green functions G(z2) = (X1X∗ 2−z2)−1,G(z2) = (X∗ 2X1−z2)−1. 26 PROPOSITION 6.1. With high probability, X0−zis invertible uniformly in \|z\| ≥p ρ(V) +δ′for some δ′>0, and in addition (6.9) \|\|(X0−z)−1\|\|op<1 δ,\|\|G(z2)\|\|op<1 δ,\|\|G(z2)\|\|op<1 δ hold for some constant δ >0, uniformly in \|z\| ≥p ρ(V) +δ′. Here, we first give the proof of Proposition 6.1, assuming (6.8) is satisfied. Since the system of equations ( 6.5) is scaling invariant, in the following we may assume ρ(V) = 1 . PROOF OF PROPOSITION 6.1. Since one can find δ′>0such that (6.8) holds, then z /∈ eDδ′. This implies, by a slight abuse of notations, dist(0,suppρz)> δ′. Further, Theorem 4.7 in [1] tells us that uniformly in z, no eigenvalue of Hzis away from suppρzwith high probability. Specifically, for a fixed z, there exist constants δ0,C > 0such that P Spec(Hz)⊂ {x∈R:dist(x,suppρz)≤N−δ0} ≥1−C N. Hence, there exists δ >0such that the smallest singular value of X0−zsatisfies P(σmin(X0−z)> δ)≥1−C N. Then with high probability, X0−zis invertible and (6.10) \|\|(X0−z)−1\|\|op<1 δ. According to Schur’s complement (X0−z)−1= (−z+1 zX1X∗ 2)−1 (−z+1 zX∗ 2X1)−1 . zG(z2)is a submatrix of X0−z, then by ( 6.10), for\|z\|2>1 \|\|G(z2)\|\|op<1 δ. The uniformity of the estimates follow simply by applying Neumann expansion. For in- stance, for sufficiently large z, say,\|z\| ≥NCfor some large constant C, one can expand G(z)around −z−1, by applying a crude order 1upper bound of the operator norm X1X∗ 2. For thosep ρ(V) +δ′≤ \|z\| ≤NC, we can apply a standard ϵ-net argument. We can find anϵ-net of this domain with cardinality NO(C)such that for each zin this domain, one can find an z′in the ϵ-net, so that \|z−z′\| ≤N−C. By the definition of stochastic dominance in Definition 1.3, one can readily conclude that the estimates in (6.9) hold uniformly on the ϵ-net, with high probability. Then for any other zsatisfyingp ρ(V) +δ′≤ \|z\| ≤NC, we can expand G(z)around G(z′)using Neumann expansion, where z′is a point in the ϵ-net and \|z−z′\| ≤N−2C. Then by	https://arxiv.org/abs/2504.19450v1
the boundedness of the operator norm G(z′), one can conclude the proof of the uniformity for all \|z\| ≥p ρ(V) +δ′. In the sequel, we prove (6.8), which follows from the lemma below, according to (6.4). SIGNAL DETECTION VIA ASYMMETRIZATION 27 LEMMA 6.2. The solution of (6.5) satisfies (6.11) ⟨u(η)⟩=⟨v(η)⟩ for all η >0, where ⟨a(η)⟩= (n+p)−1Pn+p i=1ai(η)fora=u,v. Uniformly in 0< η≤1 and\|z\|2>1 +δ′, we have the estimate (6.12) u(η)∼v(η)∼η \|z\|2−1 +η2/3. Here we follow the proof strategy of Proposition 3.2 in [2]. The main difference is that our variance matrix Vhas zero blocks and the entries of Vdo not satisfy the flatness assump- tion, although the off-diagonal blocks do satisfy the flatness assumption; see (1.1). Necessary modifications will be made in the following proof. PROOF OF LEMMA 6.2. First, multiplying on both sides of two equations in ( 6.5) byη+ V∗uandη+Vvrespectively, we get (6.13)u η+V∗u=v η+Vv which leads to (6.14) 0 =η(u−v) + (uVv−vV∗u) Taking average on both sides and using the fact that ⟨uVv⟩=⟨vV∗u⟩, we obtain ( 6.11). The matrix Vdoes not satisfy the flatness assumption, namely we cannot deduce the esti- mate immediately from assumption (1.1) on Sthat (6.15) Vv∼ ⟨v(η)⟩,Vu∼ ⟨u(η)⟩ But we can still make use of the block structure of Vto prove (6.15). Note that by the defini- tion in (1.7) and (6.16) Vv=Tv2 T∗v1 , we immediately get from (1.1) the estimate (6.17) ⟨v1⟩ ∼T∗v1,⟨v2⟩ ∼Tv2,⟨u1⟩ ∼T∗u1,⟨u2⟩ ∼Tu2 Now together with ( 6.16), (6.17) and the fact that ⟨u⟩=p n+p⟨u1⟩+n n+p⟨u2⟩,⟨v⟩=p n+p⟨v1⟩+n n+p⟨v2⟩, (6.18) in order to prove ( 6.15), it suffices to show (6.19) ⟨u1⟩ ∼ ⟨u2⟩,⟨v1⟩ ∼ ⟨v2⟩ We first need to show an auxiliary bound for ⟨u⟩and⟨v⟩ (6.20) η≲⟨u⟩=⟨v⟩≲1 From the first equation in ( 6.6) and ( 6.17) we have (6.21) u1=η+Tu2 (η+Tu2)(η+Tv2) +\|z\|2∼η+⟨u2⟩ (η+⟨u2⟩)(η+⟨v2⟩) +\|z\|2 Suppose ⟨u⟩=⟨v⟩≲η, then ⟨u2⟩,⟨v2⟩≲ηfollows immediately from ( 6.18). Then ( 6.21) gives⟨u1⟩ ∼η. The lower bound follows. 28 To show the upper bound for ⟨u⟩and⟨v⟩, from ( 6.6) one can obtain u1 η+Tu2=v1 η+Tv2,u2 η+T∗u1=v2 η+T∗v1. Multiplying T,T∗on both sides of the above two equations respectively, and using ( 6.17) gives ⟨u1⟩ η+⟨u2⟩∼⟨v1⟩ η+⟨v2⟩,⟨u2⟩ η+⟨u1⟩∼⟨v2⟩ η+⟨v1⟩. (6.22) Using the lower bound on ⟨u⟩and (6.18), we may assume ⟨u2⟩≳η. In case ⟨v2⟩≳ηalso holds, the first estimation becomes (6.23)⟨u1⟩ ⟨u2⟩∼⟨v1⟩ ⟨v2⟩ If⟨v2⟩≲η, we can easily get from (6.21) that ⟨u1⟩≳η. This together with the second equa- tion in (6.22) that (6.23) is still valid. It then follows from (6.11), (6.18) and (6.23) that (6.24) ⟨u1⟩ ∼ ⟨v1⟩,⟨u2⟩ ∼ ⟨v2⟩. Now, from the first equation in ( 6.6), we know (6.25) 1 =ηu1+u1Tv2+\|z\|2u1 η+Tu2≥u1Tv2 Taking average gives (6.26) 1≥ ⟨u1Tv2⟩≳⟨u1⟩⟨v2⟩ Similarly, using the second equation in ( 6.6),1≳⟨u2⟩⟨v1⟩. If⟨u1⟩,⟨v2⟩≲1, together with (6.18), (6.24), the upper bound follows. Otherwise, suppose ⟨u2⟩is of order greater than 1, then ⟨u1⟩ ∼ ⟨v1⟩is of order smaller than 1. However, the second equation in ( 6.6) gives ⟨u2⟩≲η+⟨u1⟩, which leads to a contradiction. Hence, ( 6.20) is proved. Using ( 6.20), (6.21) and ( 6.24) and supposing ⟨u2⟩≳η, we obtain (6.27) u1∼1 η+⟨u2⟩+\|z\|2 η+⟨u2⟩∼ ⟨u2⟩. Hence, ( 6.19) follows by taking average. Consequently,	https://arxiv.org/abs/2504.19450v1
we have (6.15). With ( 6.15), the remaining proof can be done similarly to that in [2]. Using the first equa- tion of ( 6.5) gives (6.28) η=u(η+Vv)(η+V∗u) +\|z\|2u− V∗u. By the Perron-Frobenius theorem, there exists a vector ϱ∈Rn+p +such that (6.29) Vϱ=ϱ,⟨ϱ⟩= 1,ϱ∼1 Note that Vhas zero blocks and indeed we apply the Perron-Frobenius theorem to TT∗and T∗Tto get ϱ∈Rn+p +. Taking scalar product of ( 6.28) with ϱ, we get (6.30) η=⟨ϱu(η+Vv)(η+V∗u)⟩+ (\|z\|2−1)⟨ϱu⟩ ∼ ⟨u⟩3+ (\|z\|2−1)⟨u⟩ where in the last step we also use ( 6.11) and ( 6.15). Now one can conclude ( 6.12) foru(η), and similarly for v(η). It further implies (6.8). Then we can conclude the proof of Proposition 6.1. SIGNAL DETECTION VIA ASYMMETRIZATION 29 Next, we extend the conclusions in Proposition 6.1 from the matrix X0toX. We have the following proposition. PROPOSITION 6.3. With high probability, X − zis invertible uniformly in all \|z\| ≥p ρ(V) +δ′for some δ′>0. PROOF . We first show the proof for the invertibility for one fixed zwhich is larger thanp ρ(V) +δ′in magnitude. Later, we will prove the uniformity. From Proposition 6.1, we know (6.31) det(X0−z)̸= 0 with high probability. It suffices to show that det(X −z)̸= 0. Recall the definition from (1.3) X= ΣX1 (ΣX2)∗ We use the following basic identity of the determinant of a block matrix (6.32) det A B C D = det( A − BD−1C)·det(D) which holds when Dis invertible. One can then easily check that det(X −z) = (−1)pdet(Σ X1X∗ 2Σ∗−z2) = (−1)pdet(ΣΣ∗)·det(X1X∗ 2−z2(Σ∗Σ)−1) (6.33) Recall from (3.6) that Γis a low rank diagonal matrix which satisfies det(I−Γ) = 1/det(Σ∗Σ). Since (6.31) implies that det(X1X∗ 2−z2)̸= 0, we have det(X1X∗ 2−z2(Σ∗Σ)−1) = det( X1X∗ 2−z2+z2LΓL∗) = det( X1X∗ 2−z2)·det(I+z2ΓL∗(X1X∗ 2−z2)−1L) (6.34) By our assumption, 1−Γiiis no smaller than λ−1 max(Σ∗Σ). Further using the fact that λmax(Σ∗Σ)·q−1 n=o(1), we have that 1−Γiiis of order greater than q−1 nfor all i. Moreover, using the fact that Γis of low rank and diagonal, we have for some fixed constant k det(I+z2ΓL∗(X1X∗ 2−z2)−1L) = (1 + o(1))·kY i=1(1 +z2Γiic∗ i(X1X2−z2)−1ci) = (1 + o(1))·kY i=1(1 +z2Γii(−1 z2+O≺(q−1 n))) = (1 + o(1))·kY i=1(1−Γii+O≺(q−1 n)) (6.35) where the first two steps follow from Proposition 3.2, and especially in the first step we used the fact that all the off-diagonal entries are of order q−1 n≪1−Γii. Now together with (6.33), (6.34) and (6.35), det(X −z)̸= 0holds with high probability, for a fixed z. Next, we show that the fact det(X −z)̸= 0 holds uniformly in \|z\| ≥p ρ(V) +δ′, with high probability. It suffices to have Proposition 3.2 uniformly in z. Again, the uniformity of the estimates in Proposition 3.2 follows simply by applying Neumann expansion. For instance, for sufficiently large z, say,\|z\| ≥NCfor some large constant C, one can expand G(z)around −z−1, and 30 apply a crude upper bound of the operator norm X1X∗ 2, to see that Proposition 3.2 is true uniformly in \|z\| ≥NC, with high probability. For thosep ρ(V) +δ′≤ \|z\| ≤NC, we can apply a standard ϵ-net argument. We can find an ϵ-net of this	https://arxiv.org/abs/2504.19450v1
domain with cardinality NO(C) such that for each zin this domain, one can find an z′in the ϵ-net, so that \|z−z′\| ≤N−C. By the definition of stochastic dominance in Definition 1.3, one can readily conclude from Proposition 3.2 that the estimates therein holds uniformly on the ϵ-net, with high probability. Then for any other zsatisfyingp ρ(V) +δ′≤ \|z\| ≤NC, we can expand G(z)around G(z′) using Neumann expansion, where z′is a point in the ϵ-net and \|z−z′\| ≤N−2C. Then by the boundedness of the operator norm G(z′)from Proposition 6.1, one can conclude the proof of the uniformity of the estimate in Proposition 3.2 for all \|z\| ≥p ρ(V) +δ′, which holds with high probability. PROOF OF PROPOSITION 3.1. With Propositions 6.1 and 6.3, we can conclude the proof of Proposition 3.1. 7. A Priori bound for Green function quadratic forms: Proof of Proposition 3.2. PROOF OF PROPOSITION 3.2. For brevity, in the sequel, we show the proof of the first bound in (3.5). The other terms can be bounded in a similar manner and thus we omit the details. Recall the definition of qnfrom (3.4). We denote by P:=zqnu∗GvχG=qnu∗X1X∗ 2Gvχ G, where χGis defined in (5.2). We also refer to the discussion in (5.2)-(5.7) regarding the χG factor. Therefore, in the sequel, we may simply regard χGas1and neglect all the terms involving its derivatives. Notice that from the definition in (3.3) we have the trivial bound \|P\|=\|qnu∗v+zqnu∗Gv\|χG≺qn (7.1) and also the deterministic bound \|P\| ≤ CKn for some constant C >0; see (5.3). We aim for a recursive moment estimate for E(P)2k. By cumulant expansion formula [40, Lemma 1.3], we have E(P)2k=qnX ijEh x1,ij(X∗ 2Gvu∗)jiχG(P)2k−1i =qnmX α=1X ijκα+1(x1,ij) α!E∂α 1,ijh (X∗ 2Gvu∗)jiχG(P)2k−1i +R, (7.2) where we again use Rto denote the remainder term, with certain abuse of notation. Specifi- cally, here Ris given by \|R\| ≤ CE\|x1,ij\|m+2E sup \|x1,ij\|≤c ∂m+1 1,ijh (X∗ 2Gvu∗)jiχG(P)2k−1i +CE \|x1,ij\|m+21(\|x1,ij\|> c) E sup x1,ij∈R ∂m+1 1,ijh (X∗ 2Gvu∗)jiχG(P)2k−1i . (7.3) As the derivative can only generate matrix entries which are O≺(1), and there are totally 2k−1qn-factors from P2k−1, we can trivially bound ∂m+1 1,ijh (X∗ 2Gvu∗)jiχG(P2k−1) ≺q2k−1 n. SIGNAL DETECTION VIA ASYMMETRIZATION 31 In order to bound the remainder terms in the cumulant expansion, we also need to have the above bound when we take supremum over one Xentry. This technical problem can be handled by simply truncating the matrix entries at n−1/2+εat the beginning. We omit the details. Further notice that E\|x1,ij\|m+2∼n−m+2 2.Hence, it is easy to show that when mis sufficiently large, \|R\| ≺ n−C(7.4) for a constant C=C(m,ε)>0. From the first mterms in (7.2), we start from α= 1. After an elementary calculation, we arrive at qnX ijκ2(x1,ij)E∂1 1,ijh (X∗ 2Gvu∗)jiχG(P)2k−1i =qn nEh v∗G∗X2X∗ 2Guχ G· P2k−1i −zq2 n nEh v∗G∗X2X∗ 2Gv·u∗G∗uχG· P2k−2i =qn nEh O≺(1)P2k−1i +q2 n nEh O≺(1)P2k−2i Forα≥0, we first notice that each term in ∂α 1,ij(X∗ 2Gvu∗)jimust contain the factor (X∗ 2Gvu∗)ji and the other factors can be simply bounded by O≺(1). Further, we notice that each term in ∂α 1,ijPmust contain the factor u∗Gei·e∗ jX∗ 2Gv and the	https://arxiv.org/abs/2504.19450v1
other factors can be simply bounded by O≺(1). Apart from the qnfactors, the ijsum of all the derivatives can be simply bounded by X ij\|e∗ j(X∗ 2Gvu∗)ei\|\|e∗ jX∗ 2Gv\|\|u∗Gei\| ≤X i\|u∗Gei\|sX j\|e∗ j(X∗ 2Gvu∗)ei\|2sX j\|e∗ jX∗ 2Gv\|2 ≤X i\|u∗Gei\|\|u∗ei\|p v∗G∗X2X∗ 2Gvp v∗G∗X2X∗ 2Gv ≺X i\|u∗Gei\|\|u∗ei\| ≺1, where the last step follows from Cauchy Schwarz inequality. Then, if we further consider theqnfactors generated from the derivative of P, the most dangerous terms from ∂α 1,ijP2k−1 would be (∂1 1,ijP)αP2k−1−α which creates a factor qα n. The contribution of such term to (7.2) would be qn√nα+1 P2k−1−α. Putting all these estimates together, we arrive at the recursive moment estimate E\|P\|2k=2kX α=1qn√nα Eh O≺(1)P2k−αi +O(n−C). 32 Applying a Young’s inequality, we can immediately get E\|P\|2k=o(1). Due to arbitrariness of k, we obtain the first estimate in (3.5). The other terms can be esti- mated similarly. Hence, we conclude the proof of Proposition 3.2. Acknowledgement. Z. Bao is grateful to Johannes Alt and Torben Kr ¨uger for explaining the work [1] and several related works. We would also like to thank Xiucai Ding for refer- ence. Z. Bao is supported by Hong Kong RGC Grant GRF 16304724, NSFC12222121 and NSFC12271475. K. Cheong and Y . Li are supported by Hong Kong RGC Grant 16303922. REFERENCES [1] Alt, J., Erd ˝os, L., Kr ¨uger, T., Nemish, Y . (2019). Location of the spectrum of Kronecker random matrices. Annales de lI.H.P. Probabilit ´es et Statistiques, 55(2). https://doi.org/10.1214/18-AIHP894 [2] Alt, J., Erd ˝os, L., Kr ¨uger, T. (2018). Local inhomogeneous circular law. The Annals of Applied Probability, 28(1), 148203. https://doi.org/10.1214/17-AAP1302 [3] Alt J, Erd ˝os, L., Kr ¨uger, T. Spectral radius of random matrices with independent entries. Probability and Mathematical Physics. 2021 May 22;2(2):221-80. [4] Ajanki, O. H., Erd ˝os, L., Kr ¨uger, T. (2019). Stability of the matrix Dyson equation and random matrices with correlations. Probability Theory and Related Fields, 173, 293-373. [5] Auffinger A, Ben Arous G, P ´ech´e S. Poisson convergence for the largest eigenvalues of heavy tailed random matrices. InAnnales de l’IHP Probabilits et statistiques 2009 (V ol. 45, No. 3, pp. 589-610). [6] Bai Z, Yao JF. Central limit theorems for eigenvalues in a spiked population model. InAnnales de l’IHP Probabilit ´es et statistiques 2008 (V ol. 44, No. 3, pp. 447-474). [7] Bai Z, Yao J. On sample eigenvalues in a generalized spiked population model. Journal of Multivariate Analysis. 2012 Apr 1;106:167-77. [8] Bai ZD, Yin YQ. Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 1993 Jul 1;21(3):1275-94. [9] Baik J, Ben Arous G, P ´ech´e S. Phase transition of the largest eigenvalue for nonnull complex sample co- variance matrices. Ann. Probab., 33(5):16431697, 2005. [10] Baik J, Silverstein JW. Eigenvalues of large sample covariance matrices of spiked population models. Journal of multivariate analysis. 2006 Jul 1;97(6):1382-408. [11] Bao, Z., Ding, X., Wang, K. Singular vector and singular subspace distribution for the matrix denoising model. Ann. Statist. 49(1): 370-392 (2021) [12] Bao Z, Ding X, Wang J, Wang K. Principal components of spiked covariance matrices in the supercritical	https://arxiv.org/abs/2504.19450v1
regime. Ann. Stat., 50 (2), 1144-1169, (2022). [13] Bao Z, He Y , Yang F. Random matrix theory: Local laws and applications. Handbook of Statistics. 2024. [14] Bao, Z., Pan, G., Zhou, W. Universality for the largest eigenvalue of sample covariance matrices with general population. Ann. Statist., 43(1):382421, 2015. [15] Bao Z, Wang D. Eigenvector distribution in the critical regime of BBP transition. Probability Theory and Related Fields. 2022 Feb;182(1):399-479. [16] Belinschi S, Bordenave C, Capitaine M, Cbron G. Outlier eigenvalues for non-Hermitian polynomials in independent iid matrices and deterministic matrices. Electronic Journal of Probability. 2021;26:1-37. [17] Benaych-Georges F, Guionnet A, Maida M. Fluctuations of the extreme eigenvalues of finite rank deforma- tions of random matrices. Electron. J. Probab., 16:1621-1662, 2011. [18] Benaych-Georges F, Nadakuditi RR. The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Advances in Mathematics. 2011 May 1;227(1):494-521. [19] Benaych-Georges F, Nadakuditi RR. The singular values and vectors of low rank perturbations of large rectangular random matrices. Journal of Multivariate Analysis. 2012 Oct 1;111:120-35. [20] Bordenave, C., Capitaine, M. (2016). Outlier eigenvalues for deformed iid random matrices. Communica- tions on Pure and Applied Mathematics, 69(11), 2131-2194. [21] Bordenave C, Chafa ¨ı D, Garc ´ıa-Zelada D. Convergence of the spectral radius of a random matrix through its characteristic polynomial. Probability Theory and Related Fields. 2022 Apr 1:1-9. [22] Burgess, D.J., 2019. Spatial transcriptomics coming of age. Nature Reviews Genetics, 20(6), pp.317-317. [23] Capitaine, M. (2018). Limiting eigenvectors of outliers for spiked information-plus-noise type matrices. S´eminaire de Probabilit ´es XLIX, 119-164. SIGNAL DETECTION VIA ASYMMETRIZATION 33 [24] Capitaine M, Donati-Martin C, F ´eral D. The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations. Ann. Probab., 37(1):147, 2009. [25] Capitaine M, Donati-Martin C, F ´eral D. Central limit theorems for eigenvalues of deformations of Wigner matrices. InAnnales de l’IHP Probabilits et statistiques 2012 (V ol. 48, No. 1, pp. 107-133). [26] Capitaine M, Donati-Martin C. Spectrum of deformed random matrices and free probability. arXiv preprint arXiv:1607.05560. 2016 Jul 19. [27] Chen Y , Cheng C, Fan J. Asymmetry helps: Eigenvalue and eigenvector analyses of asymmetrically per- turbed low-rank matrices. Annals of statistics. 2021 Feb;49(1):435. [28] Choi Y , Taylor J, Tibshirani R. Selecting the number of principal components: Estimation of the true rank of a noisy matrix. The Annals of Statistics. 2017 Dec 1:2590-617. [29] Cipolloni G, Erd ˝os L, Xu Y . Universality of extremal eigenvalues of large random matrices. arXiv preprint arXiv:2312.08325. 2023 Dec 13. [30] Cochran RN, Horne FH. Statistically weighted principal component analysis of rapid scanning wavelength kinetics experiments. Analytical Chemistry. 1977 May 1;49(6):846-53 [31] Foi A. Clipped noisy images: Heteroskedastic modeling and practical denoising. Signal Processing. 2009 Dec 1;89(12):2609-29. [32] Ding, X. (2020). High dimensional deformed rectangular matrices with applications in matrix denoising. Bernoulli 26(1): 387-417 (2020). [33] Ding X, Yang F. Tracy-Widom distribution for heterogeneous Gram matrices with applications in signal detection. IEEE Transactions on Information Theory. 2022 May 20;68(10):6682-715. [34] Erd ˝os L, Kr ¨uger T, Schr ¨oder D. Random matrices with slow	https://arxiv.org/abs/2504.19450v1
correlation decay. InForum of Mathematics, Sigma 2019 Jan (V ol. 7, p. e8). Cambridge University Press. [35] Erd ˝os L., Knowles A., and Yau H.-T. Averaging fluctuations in resolvents of random band matrices. Ann. Henri Poincar ´e, 14(8):1837–1926, 2013. [36] Gavish M, Donoho DL. The optimal hard threshold for singular values is 4/√ 3. IEEE Transactions on Information Theory. 2014 Jun 30;60(8):5040-53. [37] Gavish M, Donoho DL. Optimal shrinkage of singular values. IEEE Transactions on Information Theory. 2017 Jan 17;63(4):2137-52. [38] Han Y . Finite rank perturbation of non-Hermitian random matrices: heavy tail and sparse regimes. arXiv preprint arXiv:2407.21543. 2024 Jul 31. [39] Hafemeister C, Satija R. Normalization and variance stabilization of single-cell RNA-seq data using regu- larized negative binomial regression. Genome biology. 2019 Dec 23;20(1):296. [40] He Y , Knowles A. Mesoscopic eigenvalue statistics of Wigner matrices. The Annals of Applied Probability, 27(3), 1510-1550, 2017. [41] Helton, J. W., Far, R. R., Speicher, R. (2007). Operator-valued semicircular elements: solving a quadratic matrix equation with positivity constraints. International Mathematics Research Notices, 2007(9), rnm086-rnm086. [42] Jung JH, Chung HW, Lee JO. Detection of signal in the spiked rectangular models. InInternational Confer- ence on Machine Learning 2021 Jul 1 (pp. 5158-5167). PMLR. [43] Johnstone IM, Nadler B. Roys largest root test under rank-one alternatives. Biometrika. 2017 Mar 1;104(1):181-93. [44] Khorunzhy AM, Khoruzhenko BA, Pastur LA. Asymptotic properties of large random matrices with inde- pendent entries. Journal of Mathematical Physics. 1996 Oct 1;37(10):5033-60. [45] Knowles A, Yin J. The isotropic semicircle law and deformation of Wigner matrices. Comm. Pure Appl. Math., 66(11): 16631750, 2013. [46] Knowles A, Yin J. The outliers of a deformed Wigner matrix. The Annals of Probability, 42(5): 19802031, 2014. [47] Landa B, Kluger Y . The dyson equalizer: Adaptive noise stabilization for low-rank signal detection and recovery. Information and Inference: A Journal of the IMA. 2025 Mar;14(1):iaae036. [48] Lee JO, Schnelli K. Local law and TracyWidom limit for sparse random matrices. Probability Theory and Related Fields. 2018 Jun;171:543-616. [49] Lin Z, Pan G, Zhao P, Zhou J. Asymptotic distribution of spiked eigenvalues in the large signal-plus-noise models. arXiv preprint arXiv:2401.11672. 2024 Jan 22. [50] Paul D. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica. 2007 Oct 1:1617-42. [51] Rajagopalan, A. B. (2015). Outlier eigenvalue fluctuations of perturbed iid matrices. University of Califor- nia, Los Angeles. [52] Salmon J, Harmany Z, Deledalle CA, Willett R. Poisson noise reduction with non-local PCA. Journal of mathematical imaging and vision. 2014 Feb;48:279-94. 34 [53] Soshnikov A. Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Electron. Commun. Probab. 9: 82-91 (2004) [54] Tao, T. (2013). Outliers in the spectrum of iid matrices with bounded rank perturbations. Probability Theory and Related Fields, 155(1), 231-263. [55] Wallach HM. Topic modeling: beyond bag-of-words. InProceedings of the 23rd international conference on Machine learning 2006 Jun 25 (pp. 977-984).	https://arxiv.org/abs/2504.19450v1
STOCHASTIC SUBSPACE VIA PROBABILISTIC PRINCIPAL COMPONENT ANALYSIS FOR CHARACTERIZING MODEL ERROR Akash Yadav University of Houston ayadav4@uh.eduRuda Zhang University of Houston rudaz@uh.edu ABSTRACT This paper proposes a probabilistic model of subspaces based on the probabilistic principal component analysis (PCA). Given a sample of vectors in the embedding space—commonly known as a snapshot matrix—this method uses quantities derived from the probabilistic PCA to construct distributions of the sample matrix, as well as the principal subspaces. It is applicable to projection-based reduced-order modeling methods, such as proper orthogonal decomposition and related model reduction methods. The stochastic subspace thus constructed can be used, for example, to char- acterize model-form uncertainty in computational mechanics. The proposed method has multiple desirable properties: (1) it is naturally justified by the probabilistic PCA and has analytic forms for the induced random matrix models; (2) it satisfies linear constraints, such as boundary conditions of all kinds, by default; (3) it has only one hyperparameter, which significantly simplifies training; and (4) its algorithm is very easy to implement. We demonstrate the performance of the proposed method via several numerical examples in computational mechanics and structural dynamics. Keywords Model error ·Model-form uncertainty ·Stochastic reduced-order modeling ·Probabilistic principal component analysis ·Stochastic proper orthogonal decomposition 1 Introduction Engineering systems can often be described by a set of partial differential equations, commonly referred to as governing equations. These governing equations are typically derived by making assumptions and simplifying the underlying physical process, which inevitably introduces errors. The errors are further manifested via numerical approximations, imprecise initial and boundary conditions, unknown model parameters, and errors in the differential equations themselves. Model error (also known as model discrepancy, model inadequacy, structural uncertainty, or model-form error in various contexts) is ubiquitous in computational engineering, but its probabilistic analysis has been a notoriously difficult challenge. Unlike parametric uncertainties that are associated with model parameters, model-form uncertainties concern the variability of the model itself, which are inherently nonparametric. Pernot [1]argues that model-form uncertainty should not be absorbed in parametric uncertainty as it leads to prediction bias. The analysis of model-form uncertainty can bearXiv:2504.19963v2 [cs.CE] 6 May 2025 SS-PPCA divided into two sub-problems: 1) characterization and 2) correction. Determining the predictive error of an inadequate model is categorized as characterization . Reducing the predictive error of an inadequate model via some form of adjustment is called correction , which is much more complex than characterization. The central focus of our current work is to characterize model error. There are various studies in the literature that address model error. The approaches can be divided into two categories: 1) direct representation and 2) indirect representation. The methods based on direct representation are mainly subdivided into two categories, depending on where model errors are addressed: 1) external, and 2) internal [ 2]. In external direct representation methods, model error is accounted for by adding a correction term to the model output, which is usually calibrated using a Gaussian process to match the observed data. One of the earliest studies to address model error was carried out by Kennedy and O’Hagan [3]in 2001, in which	https://arxiv.org/abs/2504.19963v2
the authors presented a Bayesian calibration technique as an improvement on the traditional techniques. Since then, the Bayesian inferential and modeling framework has been adopted and further developed by many studies [ 4–8]. KOH Bayesian calibration corrects the model output but has limited extrapolative capability to unobserved quantities [ 9]. Farrell et al. [10] present an adaptive modeling paradigm for Bayesian calibration, and model selection, where the authors developed an Occam- Plausibility algorithm to select the best model with calibration. However, if the initial set of models is invalid, then another set of models is required, which requires a lot of prior information and modeling experience. In general, methods using external direct representation tend to violate physical constraints [ 11,12], rely heavily on prior information [ 13], fail to elucidate model error and data error, and struggle with extrapolation and prediction of unobserved quantity [ 1,6,14]. To alleviate some of the drawbacks, Brynjarsdóttir and O’Hagan [13] and Plumlee [15] show that prior does play an important role in improving performance. He and Xiu [2]proposed a general physics-constrained model correction framework, ensuring that the corrected model adheres to physical constraints. Despite all the developments, methods based on external direct representation have several limitations, as listed above, mainly because model error is inherently nonparametric. Internal direct representation methods improve a model through intrusive modifications, rather than by adding an extra term to the output. Embedded model enrichment [ 14] provides a framework for extrapolative predictions using composite models. Morrison et al. [16] address model inadequacy with stochastic operators introduced as a source in the governing equations while preserving underlying physical constraints. Similarly, Portone and Moser [17] propose a stochastic operator for an unclosed dispersion term in an averaged advection–diffusion equation for model closure. However, modifying the governing equations of the model brings challenges in the computational cost and implementation. Additionally, significant prior knowledge of the system is required to alter the governing equations and achieve improvements effectively. Bhat et al. [18] address model-form error by embedding dynamic discrepancies within small-scale models, using Bayesian smoothing splines (BS-ANOV A) framework. Strong and Oakley [11] introduce a novel way to quantify model inadequacies by decomposing the model into sub-functions and addressing discrepancies in each sub-function. The authors show the application of their method in health economics studies. Sargsyan et al. [19] improve uncertainty representation and quantification in computational models by introducing a Bayesian framework with embedded model error representation using polynomial chaos expansion. Other studies have used internal model error correction for various application areas, e.g., Reynolds-averaged Navier Stokes (RANS) applications [ 20], large eddy simulation (LES) computations [ 21], molecular simulations [ 22], particle-laden flows [ 23], and chemical ignition modeling [ 24]. In general, internal direct representation methods require a deep understanding of 2 SS-PPCA the system to implement the changes necessary for improvement. This knowledge is not always available and often demands an extensive study of the system. An approach based on indirect representation is presented by Sargsyan et al. [12], where the authors aim to address model error at the source by hyperparameterizing	https://arxiv.org/abs/2504.19963v2
the parameters of assumptions. However, it is not always easy to pinpoint the source of the errors especially when errors arise from multiple sources. Furthermore, this approach does not correct for model error. Another indirect representation approach to address model error is based on the concept of stochastic reduced order modeling, originally introduced by Soize in the early 2000s [ 25–27] and reviewed in [ 28]. In their earlier approaches, the authors build the reduced-order model (ROM) via projection onto a deterministic subspace and construct stochastic ROMs by randomizing the coefficient matrices using the principle of maximum entropy. The term “stochastic reduced order model” is also used by a method developed around 2010 in a different context [ 29–31], which is essentially a discrete approximation of a probability distribution and can be constructed non-intrusively from a computational model. A recent work by Soize and Farhat [32] in model-form uncertainty merges ideas from projection- based reduced-order modeling [ 33,34] and random matrix theory [ 35,36]. In the new approach, instead of randomizing coefficient matrices, the reduced-order basis (ROB) is randomized. The key observation is that since the ROB determines the ROM, randomizing the ROB also randomizes the ROM, which can be used for efficient probabilistic analysis of model error. Using computationally inexpensive ROM instead of a full-order model makes the approach computationally tractable for uncertainty quantification tasks. The probabilistic model and estimation procedure proposed in [ 32] have been applied to various engineering problems [ 37]. Because this method can face challenges due to a large number of hyperparameters and complex implementation, later developments have aimed at reducing the number of hyperparameters and simplifying hyperparameter training [ 38]. In settings with multiple candidate models, Zhang and Guilleminot [39] proposed a different probabilistic model for ROBs, and has been used for various problems in computational mechanics [40, 41]. This paper proposes a novel framework for probabilistic modeling of principal subspaces. The stochastic ROMs constructed from such stochastic subspaces enable improved characterization of model uncertainty in computational mechanics. The main contributions of our work are as follows: 1.We introduce a new class of probabilistic models of subspaces, which is simple, has nice analytical properties and can be sampled efficiently. This is of general interest, and can be seen as a stochastic version of the proper orthogonal decomposition. 2.We use this stochastic subspace model to construct stochastic ROMs, and reduce the number of hyperparameters to a single scalar. This contrasts with existing stochastic ROM approaches, which use stochastic bases and typically involve more hyperparameters. 3.The optimization of the hyperparameter is fully automated and done efficiently by exploiting one-dimensional optimization algorithms. 4.We demonstrate in a series of numerical experiments that our method provides consistent and sharp uncertainty estimates, with low computational costs. This paper is organized as follows: A brief introduction to stochastic reduced-order modeling is presented in Section 2. The proposed stochastic subspace model, along with the algorithm, is described in Section 3. Critical information regarding the hyperparameter optimization is given in 3 SS-PPCA Section 4. Related works in the literature are reviewed in Section 5.	https://arxiv.org/abs/2504.19963v2
The accuracy and efficiency of the proposed method are validated using numerical examples in Section 6. Finally, Section 7 concludes the paper with a brief summary and potential future work. 2 Stochastic reduced-order modeling This section presents the concept of stochastic reduced-order modeling. The high-dimensional model (also known as the full-order model) is described in Section 2.1. Section 2.2 describes the reduced-order model and its formulation from a high-dimensional model. Finally, Section 2.3 presents the stochastic reduced-order model and its formulation from a reduced-order model. The presentation in this section is very general and can easily be applied to any engineering system. 2.1 High-dimensional model In general, we consider a parametric nonlinear system given by a set of ordinary differential equations (ODEs): ˙x=f(x, t;µ), (1) withx∈Rnandt∈[0,∞), and is subject to initial conditions x(0) = x0and linear constraints (e.g., boundary conditions) B⊺x= 0, where B∈Rn×nCD. These equations often come from a spatial discretization of a set of partial differential equations that govern a given physical system, with dimension n≫1. We call this the high-dimensional model (HDM). If the system is time- independent, Eq. (1) reduces to a set of algebraic equations: f(x;µ) = 0 . 2.2 Reduced-order model The Galerkin projection of the HDM onto a k-dim subspace Vof the state space Rngives a reduced-order model (ROM), which can be solved faster than the HDM. Let V∈St(n, k)be an orthonormal basis of the subspace V, then the ROM can be written as: x=Vq,˙q=V⊺f(Vq, t;µ), (2) withq∈Rkand initial conditions q(0) = V⊺x0. To satisfy the linear constraints, we must haveB⊺V= 0. For time-independent systems, Eq. (2) reduces to a set of algebraic equations: V⊺f(Vq;µ)=0. 2.3 Stochastic reduced-order model The stochastic reduced-order model (SROM) builds upon the structure of ROM with the difference that the deterministic ROB Vis replaced by its stochastic counterpart W. The change from a deterministic basis to a stochastic basis introduces randomness to the model and allows it to capture model error. The SROM can be written as: x=Wq,˙q=W⊺f(Wq, t;µ),W∼µV (3) where stochastic basis Wfollows a probability distribution µV. However, it is more natural to impose a probability distribution µVon the subspace rather than the basis, because the Galerkin projection is uniquely determined by the subspace and is invariant under changes of basis. In this work, we use the stochastic subspace model introduced in Section 3 to sample stochastic basis W and construct SROM, which is then used to characterize model error. 4 SS-PPCA 3 Stochastic subspace model This section presents the formulation of the stochastic subspace model using probabilistic principal component analysis. The derivation of a deterministic principal subspace via proper orthogonal decomposition is outlined in Section 3.1. The procedure of constructing random matrices is described in Section 3.2. The stochastic subspace model and the sampling algorithm are discussed in Section 3.3. 3.1 Subspace from the PCA: proper orthogonal decomposition Principal component analysis (PCA) is a common technique for dimension reduction. Closely related to PCA, a common way to find the ROB Vand the associated subspace Vis the proper orthogonal decomposition (POD). Following the notations in Section 2, let X=	https://arxiv.org/abs/2504.19963v2
[x1···xm]∈ Rn×mbe a sample of the state, with xi=x(ti;µi), sample mean x=1 mPm i=1xi, centered sample X0=X−x1⊺ m, and sample covariance S=1 mX0X⊺ 0. LetX0=Vrdiag(σr)W⊺ rbe a compact singular value decomposition (SVD), where σr∈Rr >0↓is in non-increasing order. The sample covariance matrix Scan be written as S=Vrdiag(σr/√m)2V⊺ r. POD takes the principal basis Vk—the leading keigenvectors of S—as the deterministic ROB. The corresponding subspace is theprincipal subspace Vk:=range (Vk). Since all state samples satisfy the linear constraints, Vk satisfies the constraints automatically. POD has several other desirable properties: it can extract coherent structures, provide error estimates, and is computationally efficient and straightforward to implement [42]. 3.2 Random matrices from the PPCA Probabilistic PCA (PPCA) [ 43] reformulates PCA as the maximum likelihood solution of a proba- bilistic latent variable model. Here, we briefly overview PPCA and introduce some related random matrix models. Assume that the data-generating process can be modeled as: x=µ+Uz+ϵ, where µ∈Rn,U∈ Rn×k,z∼Nk(0,Ik),ϵ∼Nn(0, ε2In),n∈Z>0is the ambient dimension, and k∈ {1,···, n}is the latent dimension. Then the observed data follows a Gaussian distribution x∼Nn(µ,C), where covariance matrix C=UU⊺+ε2In. Assume that we have a data sample X= [x1···xm]∈Rn×mof size m. LetS=Vdiag(λ)V⊺ be an eigenvalue decomposition (EVD) of the sample covariance matrix, where V∈O(n)is an orthogonal matrix and λ∈Rn ≥0↓is in non-increasing order. The maximum likelihood estimator of the model parameters (µ,U, ε2)are:eµ=x,eU=Vk[diag( λi−˜ε2)k i=1]1/2Q, and ˜ε2= 1 n−kPn i=k+1λi, where Vk= [v1···vk]consists of the first kcolumns of V, and Q∈O(k) is any order- korthogonal matrix. The estimated covariance matrix can thus be written as eC= Vkdiag( λi−˜ε2)k i=1V⊺ k+ ˜ε2In. We call the columns of Vkprincipal components of the data sample andλk= (λi)k i=1the corresponding principal variances . The PPCA assumes that the latent dimension kis known, and estimates the unknown data distribution x∼Nn(µ,C)by maximum likelihood as ex∼Nn(eµ,eC). Without assuming kas known, the data distribution can simply be modeled as ex∼Nn(x,S), the multivariate Gaussian distribution parameterized by the sample mean and the sample covariance. From this empirical data distribution, we can derive distributions of some related matrices, all in analytical form. 5 SS-PPCA First, the data sample matrix has a matrix-variate Gaussian distribution eX= [ex1···exm]∼ Nn,m(x;S,Im), where the columns are independent and identically distributed as the empirical data distribution exiiid∼Nn(x,S),i= 1,···, m. The orientation of the data sample matrix of size khas a matrix angular central Gaussian (MACG) distribution [44]: eVk:=π(eX0,k)∼MACG n,k(S), (4) where eX0,k∼Nn,k(0;S,Ik)is a centered k-sample matrix and π(M) :=M(M⊺M)−1/2isor- thonormalization by polar decomposition . As a mapping, π:Rn×k ∗7→St(n, k)is uniquely defined for all full column-rank n-by-kmatrices, k≤n, and takes values in the Stiefel manifold St(n, k) :={V∈Rn×k:V⊺V=Ik}. The range (i.e., column space) of the data sample matrix of sizekalso has an MACG distribution: eVk:=range (eVk)∼MACG n,k(S), (5) which is supported on the Grassmann manifold Gr(n, k)ofk-dim subspaces of the Euclidean space Rn. Its probability density function (PDF) has a unique mode—and therefore a unique global maximal point—at the principal subspace Vk, see A. This distribution is useful, for example, for modeling subspaces for parametric ROM [34]. 3.3 Stochastic subspace via the PPCA Here we propose a new class	https://arxiv.org/abs/2504.19963v2
of stochastic subspace models MACG n,k,β(S)withβ∈[k,∞). The classical MACG n,k(S)distribution is a special case with β=k. Define the principal subspace map πk(X) := range (Uk), where Ukconsists of singular vectors associated with the klargest singular values of X. As a mapping, πk:Rn×m k>7→Gr(n, k)is uniquely defined for all n-by-mmatrices whose k-th largest singular value is larger than the (k+ 1)-th, with k≤min(n, m). Using this map, the POD subspace can be written as: Vk=πk(X0). We define a stochastic subspace model: W:=πk(eX0,β)def∼MACG n,k,β(S), (6) whereeX0,β∼Nn,β(0;S,Iβ)is a centered β-sample matrix and β∈ {k, k+ 1,···} is the resample size. Considering its close connection with the POD, we may call this model a stochastic POD . Similar to MACG n,k, the unique mode and global maximal of MACG n,k,β is the principal subspace Vk. The hyperparameter βcontrols the concentration of the distribution: larger value means less variation around Vk. As with POD, the stochastic basis Wsampled from the stochastic subspace model defined in Eq. (6) satisfies the linear constraints B⊺W= 0automatically. Sampling from this model can be done very efficiently for n≫1ifkand the rank of Sare small, which is often the case in practical applications. Let S=Vrdiag(λr)V⊺ rbe a compact EVD where r=rank(S). We can show that: W=Vrπk diag(λr)1/2Zr×β , (7) where Zr×βis an r-by-βstandard Gaussian matrix. Algorithm 1 gives a procedure for sampling MACG r,k,β(S)withS= diag( s)2, which in effect implements the map πk(diag( s)Zr×β). For a gen- eral covariance matrix S, an orthonormal basis of a random subspace sampled from MACG n,k,β(S) can be obtained by left multiplying the output of Algorithm 1 with Vr. Using Eq. (7) instead of Eq. (6) reduces the computational cost for truncated SVD from O(nkβ)toO(rkβ). 6 SS-PPCA Algorithm 1 SS-PPCA : Stochastic subspace via probabilistic principal component analysis. Input: scale vector s∈Rr >0; subspace dimension k∈ {1,···, r}; resample size β∈ {k, k+ 1,···}. 1:Generate Zr×β∈Rr×βwith entries zijiid∼N(0,1) 2:M←diag(s)Zr×β 3:Truncated SVD: [Uk,dk,Vk]←svd(M, k) Output: W=Uk, an orthonormal basis of a random subspace sampled from MACG r,k,β(S)with S= diag( s)2. We can generalize the concentration hyperparameter βto real numbers in [k,∞), which leads to a continuous family of distributions MACG n,k,β(S)and can be useful when the optimal value of βis not much greater than one. In this case, we sample Z∈Rr×⌈β⌉, where ⌈β⌉denotes the smallest integer greater than or equal to β. To account for the effect of real-valued β, the weight of the final column of Zis set to β− ⌊β⌋, where ⌊β⌋is the largest integer less than or equal to β. The rest of the steps are the same as Algorithm 1. 4 Hyperparameter training To find the optimal hyperparameter β∈[k,∞), we minimize the following objective function: f(β) :=E[\|do(uL)−do(uE)\|2\|β], (8) where uEis the experimental or ground-truth observation of the output, uLis the low-fidelity prediction of the SROM, and do(u) :=∥u−uo L∥L2is the L2distance to the low-fidelity prediction uo Lof a reference model. Given the value of the hyperparameter β, the stochastic subspace model determines the stochastic ROM, which produces stochastic predictions uLthat is then summarized into the random variable do(uL). Given the experimental or ground-truth observation	https://arxiv.org/abs/2504.19963v2
uE, the objective function measures the mean squared error between do(uL)anddo(uE), which is a statistical measure of how closely the SROM resembles the ground truth. Overall, this optimization problem aims to improve the consistency of the SROM in characterizing the error of the reference model. We optimize the objective function f(β)efficiently using a one-dimensional optimization algorithm, such as golden section search and successive parabolic interpolation. Implementations of such algorithms are readily available in many programming languages, e.g., fminbnd() in Matlab, scipy.optimize.minimize_scalar() in Python, and optimize() in R. The objective function, however, is not accessible due to the expectation over SROM. To reduce computation, we ap- proximate f(β)at integer βvalues by Monte Carlo sampling, using 1,000 random samples of the SROM, and linearly interpolate f(β)between consecutive integer points. In practice, when the optimization algorithm queries f(β)at a real-valued β, its values at the two closest integer points will be computed using Monte Carlo sampling and stored. If future queries need the value of f(β) at a previously evaluated integer point, the stored value will be used to avoid re-evaluation. The optimization scheme employed in this study is very easy to implement, although it may not be the most efficient due to Monte Carlo approximation of the expectation operator. We leave improvements to the hyperparameter optimization procedure for future work. 7 SS-PPCA 5 Related works This section reviews in detail the two main works that use SROM for model error characterization. Firstly, the work by Soize and Farhat [32] is discussed in Section 5.1. In Section 5.2, the work by Zhang and Guilleminot [39] for characterizing model-form uncertainty across multiple models is presented. In Section 5.3, we distinguish the three methods based on their operational mechanisms and their dependence on available information. 5.1 Non-parametric model (NPM) The seminal work of NPM [ 32] introduced the idea of using SROM to analyze model-form uncertainty. The stochastic basis in the NPM can be written as: W=π V+PSt V(PB⊥U) ,U=sGR,G∼ GP (γ). (9) The random matrix Grequires knowledge of the discretization Dof the spatial fields, and is sampled from a Gaussian process with a hyperparameter γfor correlation length-scale. The column dependence matrix Ris an order- kupper triangular matrix with positive diagonal entries. The scale s∈[0,∞)is a non-negative number. Once Uis constructed, orthogonal projection PB⊥enforces the linear constraints, tangential projection PSt Vprojects a matrix to the tangent space of the Stiefel manifold at a reference ROB V, and orthonormalization πas in Eq. (4) gives an orthonormal basis. The hyperparameters (s, γ,R)have a dimension of k(k+ 1)/2 + 2 , and are trained by minimizing a weighted average of JmeanandJstd: the former aims at matching the SROM mean prediction to ground-truth observations O; the latter aims at matching the SROM standard deviation to a scaled difference between Oanduo L. 5.2 Riemannian stochastic model (RSM) For problems where there exist a number of physically plausible candidate models, the RSM [ 39] characterizes model-form uncertainty using SROMs constructed with the following stochastic basis: W= expSt Vn cqX i=1pilogSt V(V(i))o ,p∼Dirichlet (α), (10) where c∈[0,∞)is a scale parameter, qis the number of	https://arxiv.org/abs/2504.19963v2
models, and p= (pi)q i=1is a probability vector following the Dirichlet distribution with concentration parameters α∈Rq >0. The Riemannian logarithm logSt Vmaps a model-specific ROB V(i)to the tangent space of the Stiefel manifold at a reference ROB V. The Riemannian exponential expSt Vmaps a tangent vector at Vback to the Stiefel manifold, giving the orthonormal basis W. The scale parameter cis usually set to one or omitted in later works [ 40,41]. The hyperparameters αare trained by minimizing ∥E[logSt VW]∥2 F, to match the center of mass of the distribution to V. It is a quadratic program and can be solved efficiently. We note that the RSM cannot be applied outside multi-model settings. 5.3 Comparison of model structures To deepen understanding of the three methods, we outline their structures in Figure 1 and highlight a few key differences below. Both NPM and RSM construct probabilistic models of reduced-order bases, whereas SS-PPCA models subspaces directly. Another distinction lies in the reliance on 8 SS-PPCA NPM B,D 33// OO{X(i)}// V// µV O(i) //(s, γ,R)77 RSM B,D//{X(i)}q i=1//({V(i)}q i=1,V)// µV (c,α)77 SS-PPCA B,D//{X(i)}// (Vr,λr, k)// µV O(i) //β77 Figure 1: Dependence diagrams for (from top to bottom) the NPM and the RSM models of stochastic basis and the SS-PPCA model of stochastic subspace. The dashed lines indicate that the connection only exists when characterizing the ROM-to-HDM error. discretization information: NPM requires discretization and boundary condition data for hyper- parameter optimization and SROM sampling, while SS-PPCA and RSM do not. Finally, while RSM optimizes its hyperparameters using only the reduced bases, SS-PPCA and NPM incorporate observed quantities of interest into their objective functions. 6 Numerical experiments In this section, we evaluate the proposed method, SS-PPCA, in characterizing model error using three numerical examples. In Section 6.1, we examine a parametric linear static problem and characterize model error at unseen parameters. In Section 6.2, we demonstrate how our approach can be used to characterize the HDM error, assuming that experimental data is available. Finally, in Section 6.3, we discuss a linear dynamics problem involving a space structure. All data and code for these examples are available at https://github.com/UQUH/SS_PPCA. The performance of the SROMs is assessed using two general metrics for stochastic models: consis- tency and sharpness. By consistent UQ, we mean that the statistics on prediction uncertainty—such as standard deviation and predictive intervals (PI)—derived from the model match the prediction error on average. Without consistency, the model either under- or over-estimates its prediction error, which risks being over-confident or unnecessarily conservative. By sharp UQ, we mean the model’s PIs are narrower than alternative methods. This is essentially the probabilistic version of model accuracy and is desirable as we want to minimize error. 9 SS-PPCA X Figure 2: HDM vs ROM displacement at the test parameter 6.1 Parametric linear static problem We consider a one-dimensional parametric linear static problem with n= 1,000degrees of freedom (DoFs), governed by the system of equations: Kx(µ) =f(µ), (11) where x(µ)∈Rnis the displacement vector, and µ∈[0,1]5is a parameter vector which controls the loading conditions. The system is subject to homogeneous Dirichlet boundary conditions	https://arxiv.org/abs/2504.19963v2
at the first and the last node, i.e., x1(µ) =xn(µ) = 0 . These constraints can be compactly represented asB⊺x(µ) = 0 withB= [e1en], where eiis the i-th standard basis vector of Rn. The stiffness matrix K∈Rn×nis constructed as K=ΦΛΦ⊺, with Λ= diag(4 π2j2)n−2 j=1andΦ= [0S0]⊺. The matrix S=q 2 n−1 sinjkπ n−1j=1,...,n−2 k=1,...,n−2is the order- (n−2)type-I discrete sine transform (DST-I) matrix, scaled to be orthogonal. We can see that Λ∈R(n−2)×(n−2)andΦ∈Rn×(n−2). Therefore Khas eigenpairs (λj,ϕj)withλj= 4π2j2andϕjthej-th column of Φ. The force is given by f(µ) =∥g(µ)∥−1 ∞g(µ)with g(µ) =P6 i=2µiϕiandµi∈[0,1].This setup provides a controlled framework for studying parameter-dependent structural responses, especially in the context of model error. In this example, Kdoes not depend on µ. Hence, it can be factorized once and stored for efficient inversion. However, for a general case, the parametric HDM can be computationally expensive, especially if it needs to be evaluated multiple times. This justifies the use of ROM. We construct a ROM by evaluating the HDM at 100 parameter samples, µj∈[0,1]5, drawn via Latin Hypercube Sampling (LHS). For each sample, the solution is computed as xj=x(µj) =K−1f(µj), j= 1,···,100. These solutions are concatenated to form a snapshot matrix X, which is used to construct a deterministic ROB V=πk(X)with dimension k= 4via POD. The governing equation of the ROM is given by: xR=Vq,Kkq=V⊺f(µ),Kk=V⊺KV. (12) To evaluate the accuracy of this ROM, we define a test parameter as µtest= [1 2,1 2,1 2,1 2,1], which lies within the parameter domain but was not included in the training set. The goal is to characterize 10 SS-PPCA XDisplacement Figure 3: Linear static problem: prediction by SS-PPCA. the ROM error at this previously unseen parameter point, providing insight into the model’s generalization capabilities. Figure 2 shows the error between HDM and ROM displacement at the test parameter. To characterize the error induced by ROM, we construct SROMs by replacing the deterministic ROB Vwith SROBs W. The SROBs Ware sampled using the stochastic subspace model described in Section 3.3 and Algorithm 1. To achieve consistent UQ, the hyperparameter βis optimized by minimizing the objective function given in Eq. (8). For this example, uErepresents the HDM displacement at training parameters, and uo Lrepresents the ROM displacement at the same parameters. The optimal value of the hyperparameter βis 20, which is optimized efficiently using the strategy given in Section 4. The hyperparameter training time for this example is 9.1 seconds. Figure 3 shows the ROM error characterization for the linear parametric system at the test parameters. It can be observed that the SROM mean displacement is very close to the ROM displacement, and the SROM 95% PI is able to capture the ROM error consistently and sharply. The coverage for this example is 99%, that is, the SROM 95% PI has captured the ROM error over most of the domain. It can also be observed that this coverage is achieved while maintaining tight bounds around the HDM displacement. 6.2 Static problem: characterizing HDM error In the previous example, we demonstrated the use of SROM to characterize the model truncation error induced by ROM in a	https://arxiv.org/abs/2504.19963v2
linear parametric system. However, the utility of SROM is not limited to characterizing ROM error. It can also be employed to characterize discrepancies between HDM and experimental data, a crucial task in model validation. Here we characterize HDM error via the SROM approach in a linear static problem. The synthetic experimental data is generated by solving the system: KExE=f, (13) withn(model dimension) = 1,000,f=P5 i=20.5ϕi+ 1ϕ6, and homogeneous Dirichlet boundary conditions B⊺xE= 0where B= [e1en]. The stiffness matrix is defined as KE=K+Kϵ, where Kis formulated the same as in the previous example and Kϵis a perturbation matrix designed to induce a 5% change in the Euclidean norm of KE. In addition to the model error, we simulate 11 SS-PPCA 0 0.2 0.4 0.6 0.8 1 X-1012Displacement#10!4 GroundTruth GroundTruth(NoNoise) HDM SROMmean SROM95%PI Figure 4: Linear static problem with experimental data: SS-PPCA prediction. measurement error by adding 5% noise to the experimental displacement xE. Hence, the observed experimental data follow: xobs=xE+η, where η∼N(0, σ2I)with variance chosen to achieve the specified noise level. Experimental data can be very expensive to acquire. Therefore, we limit ourselves to experimental data from a sparse subset of spatial locations. These sparse measurements are used to characterize HDM error via the SROM approach. The computational HDM is defined as: Kx H=f, (14) with the same Kandfas above. However, the HDM produces biased predictions because the true system differs from this model. To capture this error, we perturb the HDM so that the perturbed HDM responses can sufficiently cover the experimental data. The HDM is perturbed 100 times, and the responses are collected in a snapshot matrix. ROM of dimension k= 4is then constructed using POD, which further induces model truncation error. The ROM derived from the HDM is defined as: xR=Vq,V⊺KVq =V⊺f. (15) To characterize both the ROM and HDM error, we construct SROMs by substituting the deterministic ROBVwith the SROBs W. The SROBs Ware sampled using the stochastic subspace model described in Section 3.3 and Algorithm 1. To have consistent UQ, the hyperparameter βis optimized by minimizing the objective function given in Eq. (8). For this example, uErepresents the sparsely observable experimental displacement, and uo Ldenotes the ROM displacement at the same locations. The optimum value of the hyperparameter βis 8, which is optimized efficiently using the strategy given in Section 4. The hyperparameter training time for this example is 1.9 seconds. Figure 4 shows the HDM error characterization by the SS-PPCA method using the sparse ex- perimental observations. It can be observed that our method successfully captures the model discrepancy, yielding accurate uncertainty quantification for the HDM predictions. The SROM 95% PI captures the noiseless ground-truth data well, and the bounds are very sharp. The trained SROM model via SS-PPCA enables fast prediction with error estimates and eliminates the dependence on computationally expensive HDM. 12 SS-PPCA a b 0 20 40 60 Time (ms)-3-2-10123Force (105 lbf) Y XZ Figure 5: Dynamics problem: (a) space structure; (b) loading. 6.3 Dynamics problem: space structure In Sections 6.1 and 6.2, we demonstrated that the SS-PPCA method performs well in characterizing uncertainty	https://arxiv.org/abs/2504.19963v2
with low computational cost. However, those cases involved relatively simple linear static problems. To evaluate the robustness and scalability of the method, we compare it in a much more complex linear dynamics problem of a space structure. Because of the higher model complexity and more realistic structural dynamics, this example may better assess the efficiency and effectiveness of the method. We consider a major component of a space structure [ 45] given an impulse load in the z-direction, see Figure 5. The space structure consists of two major parts: an open upper and a lower part. A large mass (approximately 100 times heavier than the remainder of the model) is placed at the center of the upper part. This mass is connected to the sidewalls via rigid links. The lower part of the structure consists of the outer cylindrical shell and an inner shock absorption block with a hollow space to hold the essential components. The impulse load, as shown in Figure 5b, is applied at the center of the mass in the z-direction, which is transmitted to the upper part through the sidewalls and then to the lower part via the mounting pedestal. The impulse load can compromise the functionality of the essential components located in the lower part. Hence, monitoring the acceleration and velocity at the critical points of the essential parts is important for their safe operation. We take the quantity of interest (QoI) to be the x-velocity of a critical point at one of the essential components. The space structure has no boundary conditions; it can be assumed to be floating in outer space. The governing equation of the HDM of the space structure is given by: MH¨x+CH˙x+KHx=f, (16) with the initial conditions x(0) = 0 and˙x(0) = 0 . The system matrices MH,KH, and force fare exported from the finite element model in LS-DYNA and CH=βHKHwith a Raleigh damping coefficient βH= 6.366e−6. We denote the solution of the HDM as xH, which is obtained by numerically integrating Eq. (16) using the Newmark- βmethod with a time step of 5×10−2ms. A single HDM simulation takes approximately 38 minutes, making it computationally expensive, especially when multiple runs are needed. To address this challenge, we construct a ROM with dimension k= 10 via POD. The ROM derived from the HDM is defined as xR=Vq,MH,V¨q+CH,V˙q+KH,Vq=V⊺f, (17) 13 SS-PPCA 0 10 20 30 40 50 60 70 Time(ms)-8-6-4-20246VelocityinX(in/s) HDM ROM SROMmean SROM95%PI 30 35 40-202 0 5 10-5056870727476-0.500.5 Figure 6: Dynamic prediction of the SS-PPCA model: velocity. where reduced matrices are denoted with a pattern AV:=V⊺AV, so that MH,V=V⊺MHV, for example. The initial condition of ROM is given by q(0) = 0 and˙q(0) = 0 . In contrast to the high simulation time of HDM, ROM takes 0.2 seconds—roughly 11,200 times faster. However, this computational efficiency comes at the expense of reduced accuracy. Hence, error characterization of ROM is essential, which is done by SROM throughout this study. To construct the SROMs, we substitute the deterministic ROB Vby the SROBs W. For the SS-PPCA method, the SROBs Ware sampled using the stochastic	https://arxiv.org/abs/2504.19963v2
subspace model described in Section 3.3 and Algorithm 1. For this example, uEcorresponds to the velocity from the HDM at a critical structural node, and uo Lcorresponds to the velocity predicted by ROM at the same location. The optimum value of the hyperparameter βcomes out to be 39, which is optimized using the strategy given in Section 4. The hyperparameter training time for this example is 785 seconds, which is low compared to the computational cost of the HDM. This shows that our method is scalable and can provide fast and reliable predictions even in complex systems. The HDM velocity shows a high-frequency behavior, especially during the initial 10 milliseconds. The ROM with just 10 modes fails to capture this behavior. Figure 6 illustrates that the 95% PI of the SS-PPCA method is notably consistent and significantly sharp, even in the initial 10 milliseconds period (see the Figure 6 inset for [0,10]ms). The SS-PPCA method not only captures the high-frequency behavior but also maintains tight bounds around it. Additionally, the SROM mean velocity of SS-PPCA closely matches the deterministic ROM velocity, as it should be. The coverage for velocity via the SS-PPCA method is 94.83%. Figure 7 shows the x-direction acceleration at the critical node predicted by the SS-PPCA. The acceleration also exhibits a high-frequency behavior, due to the impulse loading. Despite this, our SS-PPCA method is able to produce consistent UQ with sharp bounds. It is important to note that even though velocity (the QoI for this example) is used for training the hyperparameters and 14 SS-PPCA 0 10 20 30 40 50 60 70 Time(ms)-3-2-10123AccelerationinX(104in/s2)HDM ROM SROMmean SROM95%PI 30 35 40-101 0 5 10-2026870727476-0.200.2 Figure 7: Dynamic prediction of the SS-PPCA model: acceleration. 010203040506070 Time(ms)-5051015DisplacementinX(10!3in) HDM ROM SROMmean SROM95%PI30 35 40246 0 5 10-101 68707274761414.515 Figure 8: Dynamic prediction of the SS-PPCA model: displacement. displacement is used for obtaining the deterministic ROB V, our approach enables predictions of unobserved quantities such as acceleration. The fact that acceleration has not been observed makes our result very intriguing. Furthermore, our approach enables efficient predictions of any unobserved QoIs of the whole system, a capability that many studies in the literature fail to achieve. This highlights the effectiveness and practicality of our approach to engineering applications. The coverage for acceleration via the SS-PPCA method is 85.49%, which means that the error 15 SS-PPCA 0 10 20 30 40 50 60 70 Time(ms)-8-6-4-20246VelocityinY(in/s) HDM ROM SROMmean SROM95%PI 30 35 40-4-2024 0 5 10-5056870727476-101 Figure 9: Dynamic prediction of the SS-PPCA model: velocity at a random DoF. characterization can be improved a bit. However, given that the acceleration is an unobserved quantity and exhibits a high-frequency behavior, the result is quite good. Figure 8 compares the x-direction displacement at the critical node predicted by the SS-PPCA. It can be observed that SS-PPCA displacement prediction is highly accurate: the SROM, ROM, and HDM show similar behavior, and the 95% PI is very tight. The coverage for displacement via the SS-PPCA method is 96.78 %, which means that the displacement prediction is consistent and highly accurate. All	https://arxiv.org/abs/2504.19963v2
the quantities compared in Figures 6 to 8, whether observed or unobserved, are from the same node of the space structure. However, we may need to make a prediction at a different node of the system, or we may be interested in the general behavior of the system. Figure 9 shows the prediction of the SS-PPCA method for the y-direction velocity at a random node; the result is still consistent and very sharp. This shows that the SS-PPCA methods allow us to make predictions about the entire structure just by training on the observation data of a single DoF. In summary, the low computational cost of hyperparameter optimization of SS-PPCA, combined with the low computational cost of ROM, can accelerate the prediction of complex engineering phenomena. Furthermore, SS-PPCA offers a fully automated, efficient, and principled training workflow with minimal user input. This makes it well suited for practical applications. One issue with methods based on SROM is that they cannot fully eliminate model error as they rely on a ROM. This issue can be addressed by increasing the number of reduced bases or incorporating model closure terms. A study on model closure for ROMs is presented by Ahmed et al. [46]. 16 SS-PPCA 7 Conclusion We introduced a novel stochastic subspace model, which is used to characterize model error in the framework of stochastic reduced-order models. It is simple, efficient, easy to implement, and well-supported by analytical results. Our SS-PPCA model has only one hyperparameter, which is optimized systematically to improve consistency. Through various numerical examples, we reveal the characteristics of this model, show its flexibility across single and multiple model cases, establish its consistency, and quantify its remarkable sharpness. It opens up a promising path to the challenging problem of model-form uncertainty. Across all examples, the stochastic subspace model effectively captures various forms of model error, including parametric uncertainty, ROM error, and HDM errors informed by noisy experimental data. Although the examples used in this study are linear systems, our method is generally applicable for characterizing model error in nonlinear systems. However, in scenarios with large errors, mere characterization may not suffice. In such cases, future work will focus on developing practical and efficient techniques for model error correction. Acknowledgments The authors thank Johann Guilleminot and Christian Soize for invaluable discussions on this topic. This work was supported by the University of Houston. 17 SS-PPCA A Mode of MACG n,k(Σ) In this section, we establish the mode of the MACG probability distribution on Grassmann manifolds. To the best of our knowledge, this is the first appearance of such results. The matrix angular central Gaussian distribution MACG n,k(Σ)on the Grassmann manifold Gr(n, k) [44,47] is the probability distribution with the probability density function (PDF) p(XX⊺;Σ) = z−1\|X⊺Σ−1X\|−n/2with respect to the normalized invariant measure of Gr(n, k). Here Σ>0is an order- nsymmetric positive definite matrix, a k-dim subspace X=range (X)is represented by the orthogonal projection matrix XX⊺where X∈St(n, k)is an orthonormal basis, and the normalizing constant z=\|Σ\|k/2where \| · \|denotes the matrix determinant. In the following, let Σhave eigenvalues λ1≥ ··· ≥ λn>0and	https://arxiv.org/abs/2504.19963v2
corresponding eigenvectors v1,···,vnthat form an orthonormal basis of Rn. Theorem 1. The PDF of the MACG n,k(Σ)distribution on Gr(n, k)is maximized at any k-dim principal subspace of Σ. Ifλk> λ k+1, then the principal subspace Vk=range (Vk)with Vk= [v1···vk]is the unique mode of the distribution. Proof. Let us first consider a special case when Σis the diagonal matrix Λ= diag( λ1,···, λn). Let subspace X∼MACG n,k(Λ)and have an orthonormal basis X∈St(n, k). The PDF can be written as: p(X) =\|Λ\|−k/2\|X⊺Λ−1X\|−n/2= nY i=1λi!−k/2 \|X⊺Λ−1X\|−n/2. (18) We see that the PDF is a smooth function of X, and we first characterize its critical points. Taking the logarithm of the above equation, we get: logp(X) =−n 2log\|X⊺Λ−1X\| −k 2nX i=1logλi. (19) The second termk 2Pn i=1logλiis a constant, so the critical points of p(X)are the critical points oflog\|X⊺Λ−1X\|, which can be computed by setting tr(S−1∂S) = 0 where S=X⊺Λ−1X. The partial derivative of X⊺Λ−1Xcan be written as ∂(X⊺Λ−1X) = (∂X)⊺Λ−1X+X⊺Λ−1(∂X). The differential ∂X∈TXSt(n, k), meaning that it lies in the tangent space of the Stifel manifold St(n, k) at the point X, which must satisfy (∂X)⊺X+X⊺(∂X) = 0 . In other words, (∂X)⊺Xmust be antisymmetric, which allows us to write ∂Xin a projective form: ∂X=M−Xsym(X⊺M)where M∈Rn×kand sym (A) = (A+A⊺)/2. Now we have: tr(S−1∂S) =tr{(X⊺Λ−1X)−1[(∂X)⊺Λ−1X+X⊺Λ−1(∂X)]} =tr{(∂X)⊺Λ−1X(X⊺Λ−1X)−1+ (X⊺Λ−1X)−1X⊺Λ−1(∂X)} = 2tr{(X⊺Λ−1X)−1X⊺Λ−1(∂X)} = 2tr{(X⊺Λ−1X)−1X⊺Λ−1[M−Xsym(X⊺M)]} = 2tr{(X⊺Λ−1X)−1X⊺Λ−1M−sym(X⊺M)} = 2tr{(X⊺Λ−1X)−1X⊺Λ−1M−X⊺M} = 2sum{[(X⊺Λ−1X)−1X⊺Λ−1−X⊺]◦M}.(20) With tr(S−1∂S) = 0 for all M∈Rn×k, we have (X⊺Λ−1X)−1X⊺Λ−1−X⊺= 0, that is, X= Λ−1X(X⊺Λ−1X)−1. Further simplification gives us (XX⊺)(Λ−1X) = (Λ−1X). Because XX⊺is 18 SS-PPCA the orthogonal projection onto range (X), we have range (Λ−1X)⊆range (X); since both sides of the equation are k-dim subspaces, we have range (Λ−1X) =range (X). Therefore, X=range (X) is an invariant k-subspace of Λ−1, or equivalently, an invariant k-subspace of Λ. If the eigenvalues (λi)n i=1are distinct, then Xmust contain a k-combination of the standard basis {ei}n i=1. So far we have proved that the critical points of p(X)are the invariant k-subspaces of Λ. Since the Grassmann manifold Gr(n, k)is compact and without a boundary, the smooth PDF p(X) has a global minimum and a global maximum, both of which are critical points. We define two k-dim subspaces EkandEkas the ranges of the bases Ek:= [e1···ek]andEk:= [en−k+1···en], respectively. We have: logp(Ek) =−n 2log\|E⊺ kΛ−1Ek\| −k 2nX i=1logλi =−n 2log\|diag( λ−1 1,···, λ−1 k)\| −k 2nX i=1logλi =−n 2kX i=1logλ−1 i−k 2nX i=1logλi =n 2kX i=1logλi−k 2nX i=1logλi.(21) Similarly, we have logp(Ek) =n 2Pn i=n−k+1logλi−k 2Pn i=1logλi. In general, let λa1≥ ··· ≥ λak be the eigenvalues associated with an invariant k-subspace X0, then: logp(X0) =n 2kX j=1logλaj−k 2nX i=1logλi. (22) Since λ1≥ ··· ≥ λn, we have p(Ek)≤p(X0)≤p(Ek), for all critical points X0. This means that EkandEkare the global minimal point and the global maximal point of p(X), respectively; that is,p(Ek)≤p(X)≤p(Ek). Now we show that a critical point X0cannot be a local maximum unless {λa1,···, λak}= {λ1,···, λk}. Letva1,···,vakbe eigenvectors associated with eigenvalues λa1,···, λak, respec- tively, such that they form an orthonormal basis of X0. Assume that eigenvalue λ′/∈ {λa1,···, λak} andλ′> λ ak. Letv′be an eigenvector associated with the	https://arxiv.org/abs/2504.19963v2
eigenvalue λ′. Consider the trajectory X(θ) = range (X(θ)), where X(θ) = [va1···vak−1v(θ)],v(θ) = cos( θ)vak+ sin( θ)v′, and θ∈[0,π 2]. It is a constant-speed rotation that starts from X0. We can show that: logp(X(θ)) =n 2 k−1X j=1logλaj−log λ−1 akcos2θ+λ′−1sin2θ! −k 2nX i=1logλi, (23) which increases monotonically. Therefore, X0is not a local maximum. In summary, MACG n,k(Λ)has its global maximum at Ek, or any k-dim principal subspace of Λ. These subspaces are also the modes of the distribution. If λk> λ k+1, there is a unique k-dim principal subspace Ek, which is the unique mode of the distribution. For a general Σ, it has an eigenvalue decomposition Σ=VΛV⊺where V= [v1···vk]. Given a change of basis x=Vz, the previous arguments still hold, and the theorem follows immediately. 19 SS-PPCA References [1]P. Pernot, The parameter uncertainty inflation fallacy, The Journal of Chemical Physics 147 (2017). doi:10.1063/1.4994654. [2]Y . He, D. Xiu, Numerical strategy for model correction using physical constraints, Journal of Computational Physics 313 (2016) 617–634. doi:10.1016/j.jcp.2016.02.054. [3]M. C. Kennedy, A. O’Hagan, Bayesian calibration of computer models, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63 (2001) 425–464. doi:10.1111/1467- 9868.00294. [4]D. Higdon, M. Kennedy, J. C. Cavendish, J. A. Cafeo, R. D. Ryne, Combining field data and computer simulations for calibration and prediction, SIAM Journal on Scientific Computing 26 (2004) 448–466. doi:10.1137/S1064827503426693. [5]D. Higdon, J. Gattiker, B. Williams, M. Rightley, Computer model calibration using high- dimensional output, Journal of the American Statistical Association 103 (2008) 570–583. doi:10.1198/016214507000000888. [6]M. J. Bayarri, J. O. Berger, R. Paulo, J. Sacks, J. A. Cafeo, J. Cavendish, C.-H. Lin, J. Tu, A framework for validation of computer models, Technometrics 49 (2007) 138–154. doi:10.1198/004017007000000092. [7]P. Z. G. Qian, C. F. J. Wu, Bayesian hierarchical modeling for integrating low-accuracy and high-accuracy experiments, Technometrics 50 (2008) 192–204. doi:10.1198/004017008000000082. [8]R. B. Gramacy, H. K. H. Lee, Bayesian treed gaussian process models with an application to computer modeling, Journal of the American Statistical Association 103 (2008) 1119–1130. doi:10.1198/016214508000000689. [9]K. A. Maupin, L. P. Swiler, Model discrepancy calibration across experimental settings, Reli- ability Engineering and System Safety 200 (2020) 106818. doi:10.1016/j.ress.2020.106818. [10] K. Farrell, J. T. Oden, D. Faghihi, A bayesian framework for adaptive selection, calibration, and validation of coarse-grained models of atomistic systems, Journal of Computational Physics 295 (2015) 189–208. doi:10.1016/j.jcp.2015.03.071. [11] M. Strong, J. E. Oakley, When is a model good enough? deriving the expected value of model improvement via specifying internal model discrepancies, SIAM/ASA Journal on Uncertainty Quantification 2 (2014) 106–125. doi:10.1137/120889563. [12] K. Sargsyan, H. N. Najm, R. Ghanem, On the statistical calibration of physical models, International Journal of Chemical Kinetics 47 (2015) 246–276. doi:10.1002/kin.20906. [13] J. Brynjarsdóttir, A. O’Hagan, Learning about physical parameters: the importance of model discrepancy, Inverse Problems 30 (2014) 114007. doi:10.1088/0266-5611/30/11/114007. [14] T. A. Oliver, G. Terejanu, C. S. Simmons, R. D. Moser, Validating predictions of unobserved quantities, Computer Methods in Applied Mechanics and Engineering 283 (2015) 1310–1335. doi:10.1016/j.cma.2014.08.023. [15] M. Plumlee, Bayesian calibration of inexact computer models, Journal of the American Statistical Association 112 (2017) 1274–1285. doi:10.1080/01621459.2016.1211016. 20 SS-PPCA [16] R. E. Morrison,	https://arxiv.org/abs/2504.19963v2
T. A. Oliver, R. D. Moser, Representing model inadequacy: A stochastic operator approach, SIAM/ASA Journal on Uncertainty Quantification 6 (2018) 457–496. doi:10.1137/16M1106419. [17] T. Portone, R. D. Moser, Bayesian inference of an uncertain generalized diffu- sion operator, SIAM/ASA Journal on Uncertainty Quantification 10 (2022) 151–178. doi:10.1137/21M141659X. [18] K. S. Bhat, D. S. Mebane, P. Mahapatra, C. B. Storlie, Upscaling uncertainty with dynamic discrepancy for a multi-scale carbon capture system, Journal of the American Statistical Association 112 (2017) 1453–1467. doi:10.1080/01621459.2017.1295863. [19] K. Sargsyan, X. Huan, H. N. Najm, Embedded model error representation for Bayesian model calibration, International Journal for Uncertainty Quantification 9 (2019) 365–394. [20] T. A. Oliver, R. D. Moser, Bayesian uncertainty quantification applied to rans turbulence models, Journal of Physics: Conference Series 318 (2011) 042032. doi:10.1088/1742- 6596/318/4/042032. [21] X. Huan, C. Safta, K. Sargsyan, G. Geraci, M. S. Eldred, Z. Vane, G. Lacaze, J. C. Oefelein, H. N. Najm, Global sensitivity analysis and quantification of model error for large eddy simulation in scramjet design, in: 19th AIAA Non-Deterministic Approaches Conference, American Institute of Aeronautics and Astronautics, 2017. doi:10.2514/6.2017-1089. [22] P. Pernot, F. Cailliez, A critical review of statistical calibration/prediction models han- dling data inconsistency and model inadequacy, AIChE Journal 63 (2017) 4642–4665. doi:10.1002/aic.15781. [23] S. Zio, H. F. da Costa, G. M. Guerra, P. L. Paraizo, J. J. Camata, R. N. Elias, A. L. Coutinho, F. A. Rochinha, Bayesian assessment of uncertainty in viscosity closure models for turbidity currents computations, Computer Methods in Applied Mechanics and Engineering 342 (2018) 653–673. doi:10.1016/j.cma.2018.08.023. [24] L. Hakim, G. Lacaze, M. Khalil, K. Sargsyan, H. Najm, J. Oefelein, Probabilistic parameter estimation in a 2-step chemical kinetics model forn-dodecane jet autoignition, Combustion Theory and Modelling 22 (2018) 446–466. doi:10.1080/13647830.2017.1403653. [25] C. Soize, A nonparametric model of random uncertainties for reduced matrix models in struc- tural dynamics, Probabilistic Engineering Mechanics 15 (2000) 277–294. doi:10.1016/S0266- 8920(99)00028-4. [26] C. Soize, Maximum entropy approach for modeling random uncertainties in transient elastodynamics, Journal of the Acoustical Society of America 109 (2001) 1979–1996. doi:10.1121/1.1360716. [27] C. Soize, Random matrix theory for modeling uncertainties in computational mechan- ics, Computer Methods in Applied Mechanics and Engineering 194 (2005) 1333–1366. doi:10.1016/j.cma.2004.06.038. [28] C. Soize, A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics, Journal of Sound and Vibration 288 (2005) 623–652. doi:10.1016/j.jsv.2005.07.009. [29] M. Grigoriu, Reduced order models for random functions. application to stochastic problems, Applied Mathematical Modelling 33 (2009) 161–175. doi:10.1016/j.apm.2007.10.023. 21 SS-PPCA [30] M. Grigoriu, A method for solving stochastic equations by reduced order models and local approximations, Journal of Computational Physics 231 (2012) 6495–6513. doi:10.1016/j.jcp.2012.06.013. [31] J. E. Warner, M. Grigoriu, W. Aquino, Stochastic reduced order models for random vectors: Application to random eigenvalue problems, Probabilistic Engineering Mechanics 31 (2013) 1–11. doi:10.1016/j.probengmech.2012.07.001. [32] C. Soize, C. Farhat, A nonparametric probabilistic approach for quantifying uncertainties in low-dimensional and high-dimensional nonlinear models, International Journal for Numerical Methods in Engineering 109 (2017) 837–888. doi:10.1002/nme.5312. [33] P. Benner, S. Gugercin, K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Review 57 (2015) 483–531.	https://arxiv.org/abs/2504.19963v2
doi:10.1137/130932715. [34] R. Zhang, S. Mak, D. Dunson, Gaussian process subspace prediction for model reduction, SIAM Journal on Scientific Computing 44 (2022) A1428–A1449. doi:10.1137/21M1432739. [35] M. L. Mehta, Random matrices, number 142 in Pure and applied mathematics (Academic Press), 3rd ed. ed., Elsevier, Amsterdam, 2004. [36] A. Edelman, N. R. Rao, Random matrix theory, Acta Numerica 14 (2005) 233–297. doi:10.1017/s0962492904000236. [37] H. Wang, J. Guilleminot, C. Soize, Modeling uncertainties in molecular dynamics simula- tions using a stochastic reduced-order basis, Computer Methods in Applied Mechanics and Engineering 354 (2019) 37–55. doi:10.1016/j.cma.2019.05.020. [38] M.-J. Azzi, C. Ghnatios, P. Avery, C. Farhat, Acceleration of a physics-based machine learning approach for modeling and quantifying model-form uncertainties and performing model updating, Journal of Computing and Information Science in Engineering 23 (2022). doi:10.1115/1.4055546. [39] H. Zhang, J. Guilleminot, A riemannian stochastic representation for quantifying model uncertainties in molecular dynamics simulations, Computer Methods in Applied Mechanics and Engineering 403 (2023) 115702. doi:10.1016/j.cma.2022.115702. [40] H. Zhang, J. E. Dolbow, J. Guilleminot, Representing model uncertainties in brittle fracture simulations, Computer Methods in Applied Mechanics and Engineering 418 (2024) 116575. doi:10.1016/j.cma.2023.116575. [41] A. Quek, N. Ouyang, H.-M. Lin, O. Delaire, J. Guilleminot, Enhancing robustness in machine-learning-accelerated molecular dynamics: A multi-model nonparametric probabilistic approach, Mechanics of Materials 202 (2025) 105237. doi:10.1016/j.mechmat.2024.105237. [42] L. Sirovich, Turbulence and the dynamics of coherent structures part i: Coherent structures, Quarterly of Applied Mathematics 45 (1987) 561–571. doi:10.1090/qam/910462. [43] M. E. Tipping, C. M. Bishop, Probabilistic principal component analysis, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61 (1999) 611–622. doi:10.1111/1467- 9868.00196. [44] Y . Chikuse, Statistics on Special Manifolds, volume 174 of Lecture Notes in Statistics , 1st ed., Springer-Verlag, New York, 2003. doi:10.1007/978-0-387-21540-2. [45] X. Zeng, R. Ghanem, Projection pursuit adaptation on polynomial chaos expan- sions, Computer Methods in Applied Mechanics and Engineering 405 (2023) 115845. doi:10.1016/j.cma.2022.115845. 22 SS-PPCA [46] S. E. Ahmed, S. Pawar, O. San, A. Rasheed, T. Iliescu, B. R. Noack, On closures for reduced order models–a spectrum of first-principle to machine-learned avenues, Physics of Fluids 33 (2021) 091301. doi:10.1063/5.0061577. [47] Y . Chikuse, The matrix angular central gaussian distribution, Journal of Multivariate Analysis 33 (1990) 265–274. doi:10.1016/0047-259X(90)90050-R. 23	https://arxiv.org/abs/2504.19963v2
arXiv:2504.20539v1 [math.PR] 29 Apr 2025Randomstrasse101: Open Problems of 2024 Afonso S. BandeiraAnastasia Kireeva§Antoine Maillard♯Almut R¨ odder¶ May 21, 2025 Abstract Randomstrasse101 is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2024, with the goal of easing academic referencing. The blog can currently be accessed at randomstrasse101.math.ethz.ch . Introduction In this manuscript we include the blog entries in Randomstrasse101 of 2024, containing a total of fifteen Open Problems. Randomstrasse101 is a blog created and maintained by our group at the Department of Mathematics at ETH Z¨ urich1. The focus is on mathematical open problems in Probability Theory, Computation, Combinatorics, Statistics, and related topics. The goal is not to necessarily write about the most important open questions in these fields but simply to discuss open questions and conjectures2that each of us find particularly interesting or intriguing, somewhat in the same style as the first author’s –now almost a decade old– list of forty-two open problems [Ban16]. The blog was created and is currently maintained by Afonso S. Bandeira, Daniil Dmitriev, Anastasia Kireeva, Antoine Maillard, Chiara Meroni, Petar Nizic-Nikolac, Kevin Lucca, and Almut R¨ odder at ETH Z¨ urich. Each blog entry is generally written by one author and the author list of this manuscript is the union of the entry authors in it. Given the nature of this material, the writing of this manuscript is more informal than a typical academic text3. Nevertheless, we hope it is useful and inspires thought on these questions. Happy solving! Contents Entry 1 Welcome &Matrix Discrepancy (ASB) 2 Entry 2 Global Synchronization (ASB) 3 ASB: Department of Mathematics, ETH Z¨ urich, R¨ amistrasse 101, 8092 Zurich, Switzerland. bandeira@math.ethz.ch §AK: Department of Mathematics, ETH Z¨ urich, R¨ amistrasse 101, 8092 Zurich, Switzerland. anastasia.kireeva@math.ethz.ch ♯AM: ARGO Team, Inria Paris, 48 Rue Barrault, 75013 Paris, France. antoine.maillard@inria.fr . Blog entry written while at the Department of Mathematics, ETH Z¨ urich, R¨ amistrasse 101, 8092 Zurich, Switzerland. ¶AR: Department of Mathematics, ETH Z¨ urich, R¨ amistrasse 101, 8092 Zurich, Switzerland. almutmagdalena.roedder@ifor.math.ethz.ch 1The inspiration for the name of the blog will be clear to the reader after looking up the department’s address and recalling the Probability Theory focus of the blog. 2Conjectures here should be interpreted as mathematical statements that we do not know not to be true, and for which a proof or a refutal would be interesting progress. We do not necessarily have very high confidence that conjectures here are true. 3If you would like to refer to an open question or blog entry, we encourage you to refer to this manuscript instead, since it is a more stable reference. 1 Entry 3 Fitting ellipsoids to random points (AM) 4 Entry 4 Detection in Multi-Frequency synchronization (AK) 6 Entry 5 Tensor PCA and the Kikuchi algorithm (ASB) 7 Entry 6 Did just a couple of deviations suffice all along? (ASB) 9 Entry 7 Sampling from the Sherrington-Kirkpatrick Model (AR) 10 Updates 12 Entry 1 Welcome &Matrix	https://arxiv.org/abs/2504.20539v1
Discrepancy (ASB) Let me start with one of my favourite Open Problems. Conjecture 1 (Matrix Spencer) .There exists a positive universal constant Csuch that, for all positive integers n, and all choices of nself-adjoint n×nreal matrices A1, . . . , A nsatisfying, for all i∈[n],∥Ai∥ ≤1 (where ∥ · ∥denotes the spectral norm) the following holds min ε∈{±1} nX i=1εiAi ≤C√n. (1) I first learned about this question from Nikhil Srivastava at ICM 2014, who pointed me to this very nice blog post of Meka [Mek14]. To the best of my knowledge the question first appeared in a paper of Zouzias [Zou12]. I have also written about it in [Ban16] and more recently in [Ban24]. The non-commutative Khintchine inequality (or a matrix concentration inequality) gives the bound up to a√lognfactor. In the particular case of commutative matrices, one can simultaneously diagonalize the matrices and the question reduces to a vector problem (representing the diagonal of the respective matrices) where the spectral norm is replaced by the ℓ∞norm. This is precisely the setting of Joel Spencer’s celebrated “Six Standard Deviations Suffice” theorem [Spe85b] which establishes the conjecture in that setting, for C= 6. It is noteworthy that in the commutative setting a random choice does not give (1), the argument is of entropic nature. In a sense, in the other extreme situation in which the matrices behave “very non-commutatively” (or, more precisely, “freely”) we know that a random choice of signs works, from our (myself, Boedihardjo, and van Handel) recent work on matrix concentration [BBvH23]. The fact that the reason for the conjecture to be true in these two extreme situations appears to be so different (one based on entropy the other on concentration), together with the fact that we still do not know if it is true in general makes this (in my opinion) a fascinating question! In recent work, Hopkins, Raghavendra, and Shetty [HRS22] has established connections between this problem and Quantum Information, and Dadush, Jiang, and Reis [DJR22] establishes a connection with covering estimates for a certain notion of relative entropy. More recently, Bansal, Jiang, and Meka [BJM23] has proved Conjecture 1 for low-rank matrices (whose rank is nover a polylog factor) using the matrix concentration inequalities in [BBvH23] within a very nice decomposition argument. Motivated by Conjecture 1 and the suspicion that “ammount of commutativity” may play an important role in this question, we (myself, Kunisky, Mixon, and Zeng) proposed a Group theoretic special case of this conjecture in [BKMZ22]. Conjecture 2 (Group Spencer) .LetGbe a finite group of size n. Conjecture 1 holds in the particular case in which A1, . . . , A nare the n×nmatrices corresponding to the regular representation of G. 2 In [BKMZ22] we prove Conjecture 2 for simple groups, but the general case is still open. While the commutative case follows from Spencer’s theorem, giving an explicit construction is not trivial (see this Mathoverflow post: https://mathoverflow.net/q/441860 ). Entry 2 Global Synchronization (ASB) In the 17th century, Christiaan Huygens (inventor of the pendulum clock) observed that pendulum clocks have a tendency to	https://arxiv.org/abs/2504.20539v1
spontaneously synchronize when hung on the same board. This phenomenon of spontaneous synchronization of coupled oscillators has since become a central object of study in dynamical systems. We will focus here on spontaneous synchronization of ncoupled oscillators with pairwise connections given by a graph with adjacency matrix A∈Rn×n. The celebrated Kuramoto model [Kur75] models the oscillators as gradient flow on E(θ) =1 2Pn i,j=1Aij(1−cos(θi−θj)),where the parameterization θ∈[0,2π[n represents each oscillators angle in the moving frame of its natural frequency (for the sake of the sequel the derivation of this potential is not crucial you can see, e.g. [ABKS23] for more details). This motivates the following definition which is central to what follows. Definition 2.1. We say an n×nmatrix Ais globally synchronizing if the only local minima of E:Sn−1→R, parameterized by θ∈[0,2π[nand given by E(θ) =1 2nX i,j=1Aij(1−cos(θi−θj)), are the global minima corresponding to θi=c,∀i. A graph Gis said to be globally synchronizing if its adjacency matrix is globally synchronizing. Abdalla, Bandeira, Kassabov, Souza, Strogatz, and Townsend [ABKS23] recently showed that a random Erd˝ os-R´ enyi graph is globally synchronizing above the connectivity threshold with high probability (you can also see the Quanta article covering this at https://www.quantamagazine.org/new-proof-shows-that-e xpander-graphs-synchronize-20230724/ ). The same paper also shows that a uniform d-regular graph is globally synchronizing with high probability for d≥600, but it leaves open the following question. Conjecture 3 (Globally Synchronizing Regular Graphs) .A uniform random 3-regular graph is globally synchronizing with high probability (probability going to 1asn→ ∞ ). The following question is motivated by trying to understand the Burer-Monteiro algorithmic approach to community detection in the stochastic block model [Kur16]. The most recent progress is in [MABB24] which answers a high rank version of this question. Conjecture 4 (Global Synchrony with negative edges) .Given any ε >0, the n×nrandom matrix Awith zero diagonal and off-diagonal entries given by Aij= 1with probability1 2+δ −1with probability1 2−δ, withδ≥(1 +ε)q logn 2n, is globally synchronizing with high probability. Another elusive question about global synchrony is the “extremal combinatorics” version of the question. Conjecture 5 (Density threshold for Global Synchrony) .For any ε >0, there exists n >0and a graph G onnnodes such that the minimum degree of Gis at least3 4−ε nandGis not globally synchronizing. Kassabov, Strogatz, and Townsend [KST21] showed an upper bound on the3 4threshold in Conjecture 5, and the same paper contains evidence that3 4is the correct threshold (by arguing that an upper bound below 3 4would need to go beyond linear analysis of the dynamical system). 3 Entry 3 Fitting ellipsoids to random points (AM) In this entry I will discuss a seemingly simple question of high-dimensional stochastic geometry, which originated (as far as I know) in the series of works [Sau11, SCPW12, SPW13]: Forn, d≫1, when can a sequence of ni.i.d. random points in Rd, drawn from N(0,Id/d), be fitted by the boundary of a centered ellipsoid? Figure 1: Fitting Gaussian random points xi∼ N (0,Id/d) to an ellipsoid. Notice that the unit sphere itself is close to being a fit by simple concentration of measure: a random x∼ N(0,Id/d) has	https://arxiv.org/abs/2504.20539v1
(with high probability) distance O(1/√ d) to it. The covariance being I d/dis a convention which ensures E[∥xi∥2] = 1: it is clear that assuming a generic positive-definite covariance Σ ≻0 does not change this question, so we do not lose any generality with this assumption. The motivation of [Sau11, SCPW12, SPW13] for this question came from statistics: they showed that the probability of existence of an ellipsoid fit was equal to the one of the success of a canonical convex relaxation (called Minimum-Trace Factor Analysis) in a problem of low-rank matrix decomposition. Several other motivations were uncovered later on, and I recommend reading the introduction of [PTVW23] to learn more about them. Interestingly, it was soon conjectured that a sharp transition occurs in the regime n/d2→α >0, exactly at α= 1/4, see e.g. Conjecture 2 of [SPW13]. Conjecture 6 (Ellipsoid fitting) .Letn, d≥1andx1,···, xn∼ N(0,Id/d). For any ε >0: lim sup d→∞n d2≤1−ε 4⇒lim d→∞P[∃Ean ellipsoid fit to (xi)n i=1] = 1, (2) lim inf d→∞n d2≥1 +ε 4⇒lim d→∞P[∃Ean ellipsoid fit to (xi)n i=1] = 0. (3) Upper bounds – Ellipsoid fitting can be written as a semidefinite program (SDP), since any origin-centered ellipsoid satisfies E={x∈Rd:xTSx= 1}for some positive semidefinite symmetric matrix S, so: P[∃Ean ellipsoid fit to ( xi)n i=1] =P[∃S∈Rd×d:S⪰0 and xT iSxi= 1 for all i∈[n]]. (4) It’s then not hard to convince oneself that {xT iSxi= 1}n i=1is an independent system of nlinear equations onS: this already gives an upper bound ofd+1 2 ≃d2/2 to the number of points that can admit an ellipsoid fit (with high probability). As of now, this “silly” argument (which does not take into account the constraint thatS⪰0) is the best upper bound established for Conjecture 6. 4 Lower bounds – On the other hand, there has been an abundance of works to establish lower bounds [Sau11, SCPW12, SPW13, KD23, PTVW23, HKPX23, TW23, BMMP24]. Not diving into details, they essentially all rely on carefully picking a candidate solution S⋆to the linear system {xT iSxi= 1}n i=1(e.g. the least Frobenius norm solution), and establishing that S⋆⪰0 for small enough n. The best results in this vein are due to [TW23, HKPX23, BMMP24], which independently proved the following. Theorem 3.1 ([TW23, HKPX23, BMMP24]) .Letn, d≥1andx1,···, xn∼ N(0,Id/d). There is δ >0 such that: lim sup d→∞n d2≤δ⇒lim d→∞P[∃Ean ellipsoid fit to (xi)n i=1] = 1. While it is not known if these methods can be pushed all the way to δ= 1/4, it is conjectured that picking S⋆as the minimal nuclear norm solution to the linear system {xT iSxi= 1}n i=1gives this optimal value, and could be a path towards proving eq. (2) [MK24]. The transition at d2/4–Notice that the d2/4 threshold can be uncovered both by numerical simulations (solving a SDP can be done efficiently), but also by a heuristic argument: if the linear equations {Tr[SxixT i] = 1}n i=1were replaced by {Tr[SGi] = 1}n i=1with GiGaussian i.i.d. matrices, the existence of a sharp transition would be a direct consequence of Gordon’s theorem [Gor88], and the value d2/4 can even be recovered	https://arxiv.org/abs/2504.20539v1
in this case as the squared Gaussian width of the cone of positive semidefinite matrices! While this heuristic was well-known, we formalized it in [MB23], essentially by showing that the volume of the space of solutions is universal in both problems. We established from it the following sharp transition result. Theorem 3.2 ([MB23]) .For any ε, M > 0, we define EFP ε,M:∃S∈Rd×d: Sp( S)⊆[0, M]and1 nnX i=1\|xT iSxi−1\| ≤ε√ d. Notice that the original ellipsoid fitting problem of Conjecture 6 is EFP 0,∞. We prove: lim sup d→∞n d2=α <1 4⇒ ∃Mα>0 :∀ε >0,lim d→∞P[EFP ε,Mα] = 1, (5) lim inf d→∞n d2=α >1 4⇒ ∃εα>0 :∀M > 0,lim d→∞P[EFP εα,M] = 0. (6) A few remarks are in order to clarify the conclusion of Theorem 3.2. •The setting ε≪1 (but not going to 0 with d) is precisely the regime where the problem is not trivially solved by the unit sphere (i.e. S= Id), since (1 /n)Pn i=1\|xT ixi−1\|=O(1/√ d) (cf. also Fig. 1). •In the regime n/d2<1/4, Theorem 3.2 shows that there exists ellipsoids which are: ( i) well-behaved (the spectral norm of Sis bounded), and ( ii) they fit the points ( xi) up to an arbitrarily small error (but not going to 0 with d). •In the regime n/d2>1/4, Theorem 3.2 rules out any well-behaved ellipsoid (i.e. with bounded spectral norm) as a possible fit, even allowing a small fitting error. Theorem 3.2 provides the first mathematical result identifying a satisfiability transition in the ellipsoid fitting problem at the conjectured d2/4 threshold. While we were not yet able to close Conjecture 6 from it, some parts seem tantalizingly close: e.g. the non-existence of ellipsoid fits for n/d2>1/4 would now follow from showing that there is no ill-behaved ellipsoid fit (or that if there is an ill-behaved ellipsoid fit, there must also be a well-behaved one). Other works – To keep this post at a reasonable length, I have not mentioned some other recent works which are in some way related to this problem, and to which I contributed [MK24, MTM+24, ETB+24]. 5 In [MK24] we use non-rigorous tools that originate in statistical physics to extend the conjecture to other classes of random points beyond Gaussian distributions (and the conjectured satisfiability threshold can then be very different from d2/4!), but also to study the typical geometrical shape of ellipsoid fits, and even to predict analytically the performance of the methods used to prove the lower bounds discussed above (cf. the conjecture on the minimal nuclear norm solution mentioned below Theorem 3.1). Finally, in [MK24] we realized that the ideas behind Theorem 3.2 can be adapted to problems in statistical learning: precisely, we characterize optimal learning from data in a model of so-called “extensive-width” neural network, a regime which is both particularly relevant in practical applications and which had so far largely resisted to theoretical analysis. In another recent preprint [ETB+24], we build upon these ideas to study a toy model of a transformer architecture. Entry 4 Detection in Multi-Frequency synchronization (AK) In the study of average-case complexity, one	https://arxiv.org/abs/2504.20539v1
is often interested in the critical threshold of the signal-to-noise ratio at which a problem becomes tractable. In this entry, we will consider a multi-frequency synchronization problem, where one obtains noisy measurements of the relative alignments of the signal elements through multiple frequency channels. We are interested whether it becomes possible to detect the signal using a time-efficient algorithm at a lower signal-to-noise ratio compared to the single-frequency model. Let us define the synchronization problem more formally. In a general synchronization problem over a group G, one aims to recover a group-valued signal g= (g1, . . . , g n)∈Gnfrom its noisy pairwise information ongkg−1 j. One way to model such observations is through receiving a function of gkg−1 jcorrupted with additive Gaussian noise Ykj=f(gkg−1 j) +Wkj, W kj∼ N(0,1). In this entry, we will focus on a setting where measurements are available for all pairs ( j, k). Motivated by the Fourier decomposition of a non-linear objective of the non-unique games problem [BCLS20], we consider the model as receiving measurements through several frequency channels. For example, for G=U(1) ={eiφ, φ∈[0,2π)}, we consider measurements of the following form:   Y1=λ nxx∗+1√nW(1), Y2=λ nx(2) x(2)∗ +1√nW(2), ... YL=λ nx(L) x(L)∗ +1√nW(L), where W(1), . . . , W(L)are independent matrices, and x(k)= (xk 1, . . . , xk n) denotes entrywise power. This case corresponds to the angular synchronization, where the objective is to determine phases φ1, . . . , φ n∈[0,2π] from their noisy relative observation φk−φjmod 2 π, and equivalent to the synchronization over SO(2), as SO(2)∼=U(1). In the general case of compact groups, we consider the Peter-Weyl decomposition instead. Informally, in this case, the Fourier modes correspond to irreducible representations, and each pairwise measurement corresponding to an irreducible representation ρis a block matrix with blocks given by Yρ kj=λ nρ(gk)ρ g−1 j +1√nW(ρ) kjfor each ρ∈Ψ. The single-frequency model (i.e., when L= 1) reduces to the Wigner spiked matrix model. In this model, the celebrated BBP transition [BAP05] postulates that above a critical value λ≥λ∗= 1, detection is possible based on the top eigenvalue, while below this threshold, the top eigenvalue does not provide reliable 6 information on the presence of the signal as ngrows to infinity. Moreover, for a variety of dense priors on signal x, this procedure is optimal in the sense that no algorithm can detect the signal reliably below the spectral threshold, λ∗= 1, including algorithms with no constraints on the runtime [PWBM16]. Does the situation change for the multi-frequency model? For simplicity, let us assume that the signal x is sampled uniformly from L-th roots of unity, x∼Unif({e2πik/L}L−1 k=0). This corresponds to synchronization over the cyclic group Z/LZ. In this case, [PWBM16] showed that the statistical threshold is λstat= Θ(log L/L), i.e., below this threshold it is impossible to detect the signal, while above the threshold, there exists an inefficient algorithm for this task. The authors give upper and lower bounds on λstatwith exact constants, and the upper bound is lower than the spectral threshold for a sufficiently large number of	https://arxiv.org/abs/2504.20539v1
frequencies, namely, L≥11. From the computational point of view, in [KBK24], it was shown that assuming the low-degree conjecture, no polynomial-time algorithm can detect the signal below the spectral threshold, λ∗= 1, regardless of receiving additional information through additional channels. This result applies to the setting when the number of frequencies Lis constant compared to the dimension nand when the signal is sampled uniformly from SO(2) or any finite group of constant size. Combining this result with the optimality of PCA for the single-frequency model, this suggests that receiving only constant number of additional frequencies does not lower the computational threshold. This opens up an intriguing question: how much more frequencies one requires so that the detection becomes possible using an efficient algorithm below the spectral threshold? We can formulate the following conjecture. Open Problem 7. Consider synchronization model over SO(2)or synchronization model over a finite group, where the signal xis sampled uniformly over group elements. Find a scaling L=Lnof number of frequencies such that there exists a polynomial-time algorithm that can detect the signal reliably for all λ > λ comp ,Lfor some λcomp ,L<1asn→ ∞ . There is empirical evidence supporting the conjecture that, as Ldiverges, the computational threshold becomes lower than the spectral threshold: numerical simulations in [GZ19] suggest that the variant of AMP with a carefully performed initialization can surpass the threshold when the dimension nand the number of frequencies Lare comparable. In the computationally hard regime, we may also hope that there exists a sub-exponential algorithm whose runtime possibly depends on the signal strength. Such a tradeoff was observed in the sparse PCA problem [DKWB23]. For the synchronization model over finite groups, [PWBM16] proposed an algorithm for detecting the signal that works in the entire computationally hard regime. This algorithm involves maximizing a certain test function over all possible solutions g∈Gnand thus has an exponential runtime. Open Problem 8. Consider synchronization model over SO(2)or synchronization model over a finite group, where the signal xis sampled uniformly over group elements. Fix number of frequencies Lthat does not depend on n. In the computationally hard regime, i.e., when λ∈(λstat,1), does there exist a sub- exponential algorithm, i.e., an algorithm running in time expnδfor some δ <1, that can detect the signal reliably as n→ ∞ ? Entry 5 Tensor PCA and the Kikuchi algorithm (ASB) In 2014, Andrea Montanari and Emile Richard [MR14] proposed a statistical model for understanding a variant of Principal Component Analysis in Tensor data. This is currently referred as the Tensor PCA problem. We will consider symmetric version of the problem in which the signal of interest is a point in the hypercube. Given n, randλ, the goal is to estimate (or detect) an unknown “signal” x∈ {± 1}n(drawn uniformly from the hypercube), from “measurements” as follows: for i1< i2< ... < i r, Yi1,i2,...,ir=λx⊗r+Zi1,i2,...,ir where Zi1,i2,...,irare iid N(0,1) random variables (and independent from x). 7 We will consider r(and ℓ, to be introduced below) fixed and n→ ∞ , all big-O notation will be in terms ofn. Tensor PCA is believed to undergo a	https://arxiv.org/abs/2504.20539v1
statistical-to-computaional gap: without regards for computational efficiency, it is possible to estimate xforλ= Ω n−r 2+1 2 . Efficient algorithms, such as the Sum-of-Squares hierarchy, are able to solve the problem at λ=˜Ω n−r 4 , where ˜Ω hides logarithmic factors. Local methods, such as gradient descent and approximate message passing succeed at λ=˜Ω n−1 2 . For r= 2, the problem simply involves matrices, and indeed all these thresholds coincide. We point the reader to [WAM19] and references therein for more on each of these thresholds. In 2019, Alex Wein, Ahmed El Alaoui, and Cris Moore [WAM19] proposed an algorithm for this problem based on the so-called Kikuchi Free Energy, it roughly corresponds to a hierarchy of message passing algorithms. They showed that this approach achieves (up to logarithmic factors) the threshold for the sum- of-squares approach, shedding light on the gap between the message passing and sum-of-squares frameworks. We briefly describe the Kikuchi approach. We will focus on even r. There is a design parameter ℓ(with n≫ℓ≥r 2) which we will consider fixed. The Kikuchi matrix Mis then ℓ ×n ℓ matrix (with rows and columns indexed by subsets I⊂[n] of size ℓ) given by M(λ)I,J=YI∆J if \|I∆J\|=r, 0 otherwise, where I∆J= (I∪J)\(I∩J) denotes the symmetric difference. The goal is to understand when the top of the spectrum of Mreveals the spike x. It is shown in [WAM19] that this happens for λ=˜Ω n−r 4 . Since Zis rotation invariant, we can assume WLOG that x=1, the all-ones vectors. The following conjecture also appears in [Ban24] and is a reformulation of a conjecture in [WAM19]: Conjecture 9 (Kikuchi Spectral Threshold) .Given r, ℓ, n positive integers satisfying n≫ℓ≥r 2and r even (randℓwill be fixed and n→ ∞ ). For each S⊂[n]with\|S\|=rletZS∼ N(0,1), and all independent. Let λ≥0and let Mbe then ℓ ×n ℓ matrix (with rows and columns indexed by subsets I⊂[n]of size ℓ) given by M(λ)I,J=λ+ZI∆J if \|I∆J\|=r, 0 otherwise, where I∆J= (I∪J)\(I∩J)denotes the symmetric difference. Letλ♮ r,ℓdenote the threshold at which eigenvalues “pop-out” of the spectrum of M(λ): in other words λ♮ r,ℓis the real number such that, for all λ > λ♮ r,ℓ, there exists ε >0such that EλmaxM(λ)>(1 + ε+ o(1))EλmaxM(0),where o(1)is a term that goes to zero as n→ ∞ . For fixed r, we have nr 4λ♮ r,ℓ→0 asℓ→ ∞ (note that this is after one has taken n→ ∞ ). This would establish the very interesting phenomenon that there is no sharp threshold for polynomial time algorithms in Tensor PCA in the following sense: This conjecture would imply that by increasing ℓ (while remaining Θ(1), corresponding to increasing the computational cost of the algorithm while keeping it polynomial time), one would be able to decrease the critical signal-to-noise ratio in the sense of nr 4λ♮ r,ℓ getting arbitrarily close to zero. Since the bound in [WAM19] contains logarithmic factors on n(present in ˜Ω(·)), it does not allow to see this phenomenon for ℓ= Θ(1). For even rand1 2r≤ℓ <3 4rthe threshold has been characterized in my work with Giorgio Cipolloni, Dominik Schr¨ oder,	https://arxiv.org/abs/2504.20539v1
and Ramon van Handel [BCSvH24], which is a follow up to the matrix concentration inequalities (leveraging intrinsic freeness) in my work with March Boedihardjo, and Ramon van Handel [BBvH23]. 8 Entry 6 Did just a couple of deviations suffice all along? (ASB) In September 2024, at a conference on Mathematical Aspects of Learning Theory in Barcelona ( https: //www.crm.cat/mathematical-aspects-of-learning-theory/ ) Dan Spielman gave a beautiful talk on discrepancy theory and some of its applications in clinical trials. Even though this was not the precise focus of the talk, it sparked a discussion that day about lower bounds for Joel Spencer’s Six Deviations Suffice Theorem. I will describe some of the unanswered questions from this discussion. Many thanks to Dan Spielman, Tselil Schramm, Amir Yehudayoff, Petar Nizic-Nikolac, and Anastasia Kireeva with whom discussing this problem contributed to making the workshop a memorable week! Given n, a positive integer, and A, ann×nmatrix, we define the discrepancy of A, disc( A), as disc(A) = inf x∈{±1}n∥Ax∥∞. One of the motivations of this question is to understand how much smaller is inf x∈{±1}n∥Ax∥∞compared to the value, e.g. in expectation, when xis drawn uniformly from the hypercube. We point to the references in this post for a more thorough discussion of the context, motivations, and state of the art. In 1985, Joel Spencer [Spe85a] showed the cellebrated “Six Deviations Suffice” Theorem, which states that for any positive integer nand any A∈ {± 1}n×ndisc(A)≤6√n. There have since been improvements on this constant, but the question we will pursue here concerns lower bounds. When the condition of ±1 entries is replaced by an ℓ2condition on the columns, the corresponding question is the famous K´ omlos Conjecture (which coincidentally is the first open problem in my lecture notes from around a decade ago [Ban16]). Conjecture 10 (K´ omlos Conjecture) .There exists a universal constant Ksuch that, for all square matrices Awhose columns have unit ℓ2norm, disc(A)≤K. It is easy to see that the statement of this conjecture implies Spencer’s Theorem, since the columns of ann×n,±1 matrix, have ℓ2norm√n. The best current lower bound for K´ omlos is a recent construction by Tim Kunisky [Kun23] which shows a lower bound on Kof 1 +√ 2. Unfortunately, the resulting matrix is not a ±1 matrix (even after scaling), and so it does not provide a lower bound for the Spencer setting. Open Problem 11 (How many deviations are needed afterall?) .What is the value of the following quantity? sup n∈Nsup A∈{±1}n×n1√ndisc(A) It is easy to see that the matrix1 1 1−1 has discrepancy 2 =√ 2√ 2, giving a lower bound of√ 2 to this quantity. Also, Dan Spielman mentioned a numerical construction (of a specific size) that achieved a slightly larger discrepancy value. To the best of my knowledge, there is no known lower bound above 2, motivating the question: “Did just a couple of deviations suffice all along?”. Also, to the best of my knowledge, there is no known construction (or proof of existence) of an infinite family achieving a lower bound strictly larger than one. More explicitly, we formulate	https://arxiv.org/abs/2504.20539v1

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.