Discrete Gaussian Samplers over Lattices¶
This file implements oracles which return samples from a lattice following a discrete Gaussian distribution. That is, if \(σ\) is big enough relative to the provided basis, then vectors are returned with a probability proportional to \(\exp(-|x-c|_2^2/(2σ^2))\). More precisely lattice vectors in \(x ∈ Λ\) are returned with probability:
\(\exp(-|x-c|_2^2/(2σ²))/(∑_{x ∈ Λ} \exp(-|x|_2^2/(2σ²)))\)
AUTHORS:
Martin Albrecht (2014-06-28): initial version
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler
sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^10, 3.0)
sage: D(), D(), D()
((3, 0, -5, 0, -1, -3, 3, 3, -7, 2), (4, 0, 1, -2, -4, -4, 4, 0, 1, -4), (-3, 0, 4, 5, 0, 1, 3, 2, 0, -1))
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class
sage.stats.distributions.discrete_gaussian_lattice.
DiscreteGaussianDistributionLatticeSampler
(B, sigma=1, c=None, precision=None)¶ Bases:
sage.structure.sage_object.SageObject
GPV sampler for Discrete Gaussians over Lattices.
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^10, 3.0); D Discrete Gaussian sampler with σ = 3.000000, c=(0, 0, 0, 0, 0, 0, 0, 0, 0, 0) over lattice with basis [1 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 1]
We plot a histogram:
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler sage: import warnings sage: warnings.simplefilter('ignore', UserWarning) sage: D = DiscreteGaussianDistributionLatticeSampler(identity_matrix(2), 3.0) sage: S = [D() for _ in range(2^12)] sage: l = [vector(v.list() + [S.count(v)]) for v in set(S)] sage: list_plot3d(l, point_list=True, interpolation='nn') Graphics3d Object
REFERENCES:
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__init__
(B, sigma=1, c=None, precision=None)¶ Construct a discrete Gaussian sampler over the lattice \(Λ(B)\) with parameter
sigma
and center \(c\).INPUT:
B
– a basis for the lattice, one of the following:an integer matrix,
an object with a
matrix()
method, e.g.ZZ^n
, oran object where
matrix(B)
succeeds, e.g. a list of vectors.
sigma
– Gaussian parameter \(σ>0\).c
– center \(c\), any vector in \(\ZZ^n\) is supported, but \(c ∈ Λ(B)\) is faster.precision
– bit precision \(≥ 53\).
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler sage: n = 2; sigma = 3.0; m = 5000 sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^n, sigma) sage: f = D.f sage: c = D._normalisation_factor_zz(); c 56.2162803067524 sage: l = [D() for _ in range(m)] sage: v = vector(ZZ, n, (-3,-3)) sage: l.count(v), ZZ(round(m*f(v)/c)) (39, 33) sage: target = vector(ZZ, n, (0,0)) sage: l.count(target), ZZ(round(m*f(target)/c)) (116, 89) sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler sage: qf = QuadraticForm(matrix(3, [2, 1, 1, 1, 2, 1, 1, 1, 2])) sage: D = DiscreteGaussianDistributionLatticeSampler(qf, 3.0); D Discrete Gaussian sampler with σ = 3.000000, c=(0, 0, 0) over lattice with basis [2 1 1] [1 2 1] [1 1 2] sage: D() (0, 1, -1)
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__call__
()¶ Return a new sample.
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^3, 3.0, c=(1,0,0)) sage: L = [D() for _ in range(2^12)] sage: abs(mean(L).n() - D.c) 0.08303258... sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^3, 3.0, c=(1/2,0,0)) sage: L = [D() for _ in range(2^12)] # long time sage: mean(L).n() - D.c # long time (0.0607910156250000, -0.128417968750000, 0.0239257812500000)
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c
¶ Center \(c\).
Samples from this sampler will be centered at \(c\).
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^3, 3.0, c=(1,0,0)); D Discrete Gaussian sampler with σ = 3.000000, c=(1, 0, 0) over lattice with basis [1 0 0] [0 1 0] [0 0 1] sage: D.c (1, 0, 0)
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static
compute_precision
(precision, sigma)¶ Compute precision to use.
INPUT:
precision
- an integer \(> 53\) norNone
.sigma
- ifprecision
isNone
then the precision ofsigma
is used.
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler sage: DiscreteGaussianDistributionLatticeSampler.compute_precision(100, RR(3)) 100 sage: DiscreteGaussianDistributionLatticeSampler.compute_precision(100, RealField(200)(3)) 100 sage: DiscreteGaussianDistributionLatticeSampler.compute_precision(100, 3) 100 sage: DiscreteGaussianDistributionLatticeSampler.compute_precision(None, RR(3)) 53 sage: DiscreteGaussianDistributionLatticeSampler.compute_precision(None, RealField(200)(3)) 200 sage: DiscreteGaussianDistributionLatticeSampler.compute_precision(None, 3) 53
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sigma
¶ Gaussian parameter \(σ\).
Samples from this sampler will have expected norm \(\sqrt{n}σ\) where \(n\) is the dimension of the lattice.
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionLatticeSampler sage: D = DiscreteGaussianDistributionLatticeSampler(ZZ^3, 3.0, c=(1,0,0)) sage: D.sigma 3.00000000000000
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