Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: One Bit Sensing, Discrepancy, and Stolarsky Principle   A sign-linear one bit map from the $ d$-dimensional sphere $ \mathbb S ^{d}$
to the $ n$-dimensional Hamming cube $ H^n= \{ -1, +1\} ^{n}$ is given by $$ x
\to \{ \mbox{sign} (x \cdot z_j) \;:\; 1\leq j \leq n\} $$ where $ \{z_j\}
\subset \mathbb S ^{d}$. For $ 0 < \delta < 1$, we estimate $ N (d, \delta )$,
the smallest integer $ n$ so that there is a sign-linear map which has the $
\delta $-restricted isometric property, where we impose normalized geodesic
distance on $ \mathbb S ^{d}$, and Hamming metric on $ H^n$. Up to a
polylogarithmic factor, $ N (d, \delta ) \approx \delta^{-2 + \frac2{d+1}}$,
which has a dimensional correction in the power of $ \delta $. This is a
question that arises from the one bit sensing literature, and the method of
proof follows from geometric discrepancy theory. We also obtain an analogue of
the Stolarsky invariance principle for this situation, which implies that
minimizing the $L^2$ average of the embedding error is equivalent to minimizing
the discrete energy $\sum_{i,j} \big( \frac12 - d(z_i,z_j) \big)^2$, where $d$
is the normalized geodesic distance.

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The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg