<s>
In	O
linear	B-Language
algebra	I-Language
,	O
the	O
restricted	B-Algorithm
isometry	I-Algorithm
property	I-Algorithm
(	O
RIP	O
)	O
characterizes	O
matrices	B-Architecture
which	O
are	O
nearly	O
orthonormal	O
,	O
at	O
least	O
when	O
operating	O
on	O
sparse	O
vectors	O
.	O
</s>
<s>
There	O
are	O
no	O
known	O
large	O
matrices	B-Architecture
with	O
bounded	O
restricted	B-Algorithm
isometry	I-Algorithm
constants	I-Algorithm
(	O
computing	O
these	O
constants	O
is	O
strongly	O
NP-hard	O
,	O
and	O
is	O
hard	O
to	O
approximate	O
as	O
well	O
)	O
,	O
but	O
many	O
random	O
matrices	B-Architecture
have	O
been	O
shown	O
to	O
remain	O
bounded	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
has	O
been	O
shown	O
that	O
with	O
exponentially	O
high	O
probability	O
,	O
random	O
Gaussian	O
,	O
Bernoulli	O
,	O
and	O
partial	O
Fourier	O
matrices	B-Architecture
satisfy	O
the	O
RIP	O
with	O
number	O
of	O
measurements	O
nearly	O
linear	O
in	O
the	O
sparsity	O
level	O
.	O
</s>
<s>
The	O
current	O
smallest	O
upper	O
bounds	O
for	O
any	O
large	O
rectangular	O
matrices	B-Architecture
are	O
for	O
those	O
of	O
Gaussian	O
matrices	B-Architecture
.	O
</s>
<s>
Then	O
,	O
the	O
matrix	O
A	O
is	O
said	O
to	O
satisfy	O
the	O
s-restricted	O
isometry	O
property	O
with	O
restricted	B-Algorithm
isometry	I-Algorithm
constant	I-Algorithm
.	O
</s>
<s>
The	O
tightest	O
upper	O
bound	O
on	O
the	O
RIC	O
can	O
be	O
computed	O
for	O
Gaussian	O
matrices	B-Architecture
.	O
</s>
<s>
This	O
can	O
be	O
achieved	O
by	O
computing	O
the	O
exact	O
probability	O
that	O
all	O
the	O
eigenvalues	O
of	O
Wishart	O
matrices	B-Architecture
lie	O
within	O
an	O
interval	O
.	O
</s>
