Can we spot a fake?
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910154917675008 |
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| author | Mendelson, Shahar Paouris, Grigoris Vershynin, Roman |
| author_facet | Mendelson, Shahar Paouris, Grigoris Vershynin, Roman |
| contents | The problem of detecting fake data inspires the following seemingly simple mathematical question. Sample a data point $X$ from the standard normal distribution in $\mathbb{R}^n$. An adversary observes $X$ and corrupts it by adding a vector $rt$, where they can choose any vector $t$ from a fixed set $T$ of the adversary's ``tricks'', and where $r>0$ is a fixed radius. The adversary's choice of $t=t(X)$ may depend on the true data $X$. The adversary wants to hide the corruption by making the fake data $X+rt$ statistically indistinguishable from the real data $X$. What is the largest radius $r=r(T)$ for which the adversary can create an undetectable fake? We show that for highly symmetric sets $T$, the detectability radius $r(T)$ is approximately twice the scaled Gaussian width of $T$. The upper bound actually holds for arbitrary sets $T$ and generalizes to arbitrary, non-Gaussian distributions of real data $X$. The lower bound may fail for not highly symmetric $T$, but we conjecture that this problem can be solved by considering the focused version of the Gaussian width of $T$, which focuses on the most important directions of $T$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_18880 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Can we spot a fake? Mendelson, Shahar Paouris, Grigoris Vershynin, Roman Statistics Theory Probability 62G10, 60D05 G.3; I.5.1 The problem of detecting fake data inspires the following seemingly simple mathematical question. Sample a data point $X$ from the standard normal distribution in $\mathbb{R}^n$. An adversary observes $X$ and corrupts it by adding a vector $rt$, where they can choose any vector $t$ from a fixed set $T$ of the adversary's ``tricks'', and where $r>0$ is a fixed radius. The adversary's choice of $t=t(X)$ may depend on the true data $X$. The adversary wants to hide the corruption by making the fake data $X+rt$ statistically indistinguishable from the real data $X$. What is the largest radius $r=r(T)$ for which the adversary can create an undetectable fake? We show that for highly symmetric sets $T$, the detectability radius $r(T)$ is approximately twice the scaled Gaussian width of $T$. The upper bound actually holds for arbitrary sets $T$ and generalizes to arbitrary, non-Gaussian distributions of real data $X$. The lower bound may fail for not highly symmetric $T$, but we conjecture that this problem can be solved by considering the focused version of the Gaussian width of $T$, which focuses on the most important directions of $T$. |
| title | Can we spot a fake? |
| topic | Statistics Theory Probability 62G10, 60D05 G.3; I.5.1 |
| url | https://arxiv.org/abs/2410.18880 |