agd.Metrics.asym_iso

View Source

 1import numpy as np
 2from .base import Base
 3from .riemann import Riemann
 4from .isotropic import Isotropic
 5from .. import AutomaticDifferentiation as ad
 6from .. import LinearParallel as lp
 7from ..FiniteDifferences import common_field
 8
 9
10class AsymIso(Base):
11	r"""
12	A special case of AsymQuad metric taking the form
13	$
14	F(x) = \sqrt{a^2 |x|^2+sign(a) max(0,<w,x>)^2}
15	$
16	where $a$ is a field of scalars, and w a vector field.
17	Member fields and __init__ arguments : 
18	- a : an array of shape (n1,..,nk)
19	- w : an array of shape (vdim,n1,...,nk)
20	"""
21	def __init__(self,a,w):
22		a,w = (ad.asarray(e) for e in (a,w))
23		self.a,self.w =common_field((a,w),(0,1))
24
25	def norm(self,v):
26		v,a,w = common_field((ad.asarray(v),self.a,self.w),(1,0,1))
27		return np.sqrt(a**2*np.sum(v**2,axis=0) + np.sign(a)*np.maximum(lp.dot_VV(w,v),0.)**2)
28
29	def gradient(self,v):
30		v,a,w = common_field((ad.asarray(v),self.a,self.w),(1,0,1))
31		g = a**2*v + np.sign(a)*np.maximum(0.,lp.dot_VV(w,v))*w
32		return g/np.sqrt(lp.dot_VV(v,g))
33
34	def dual(self):
35		a2,s,w2 = self.a**2,np.sign(self.a),self._norm2w()
36		r2 = s*(1/a2-1/(a2+s*w2))/w2
37		return AsymIso(-1/self.a,np.sqrt(r2)*self.w)
38
39	@property
40	def vdim(self): return len(self.w)
41
42	@property
43	def shape(self): return self.a.shape
44
45	def _norm2w(self): return np.sum(self.w**2,axis=0)
46	def is_definite(self):
47		return np.where(self.a>0,1.,self.a**2-self._norm2w()) > 0
48	def anisotropy(self):
49		a2,w2 = self.a**2,self._norm2w()
50		return np.sqrt(np.where(self.a>0,1.+w2/a2,1./(1.-w2/a2)))
51	def cost_bound(self):
52		return np.where(self.a>0,np.sqrt(self.a**2+self._norm2w()),-self.a)
53
54	def rotate(self,r): return AsymIso(self.a,lp.dot_AV(r,self.w))
55	def with_cost(self,cost): return AsymIso(cost*self.a,cost*self.w)
56	def with_costs(self,costs):
57		if len(costs)>1 and not np.allclose(costs[1:],costs[0]):
58			raise ValueError("Costs must be identical along all axes for Metrics.Isotropic. Consider using Metrics.Diagonal class")
59		return self.with_cost(costs[0])
60
61	def model_HFM(self):
62		return 'AsymIso'+str(self.vdim)
63
64	@classmethod
65	def from_cast(cls,metric):
66		if isinstance(metric,cls): return metric
67		iso = Isotropic.from_cast(metric)
68		w = np.zeros_like(iso.cost,shape=(iso.vdim,)+iso.shape)
69		return cls(iso.cost,w)
70
71	def __iter__(self):
72		yield self.a
73		yield self.w
74
75	def make_proj_dual(self,**kwargs):
76		"""kwargs : passed to Riemann.make_proj_dual"""
77		vdim,a = self.vdim,self.a
78		proj_a = Isotropic(np.abs(a)).make_proj_dual()
79		eye = np.eye(vdim,like=a).reshape( (vdim,vdim)+(1,)*a.ndim )
80		proj_w=Riemann(a**2*eye+np.sign(a)*lp.outer_self(self.w)).make_proj_dual(**kwargs)
81
82		def proj(x):
83			x,w = common_field((x,self.w),depths=(1,1))
84			x_a = proj_a(x)
85			x_w = proj_w(x)
86			s = lp.dot_VV(w,x_a) > 0
87			return np.where(s[None],x_w,x_a)
88		return proj
89
90
91
92

class AsymIso(agd.Metrics.base.Base): View Source

11class AsymIso(Base):
12	r"""
13	A special case of AsymQuad metric taking the form
14	$
15	F(x) = \sqrt{a^2 |x|^2+sign(a) max(0,<w,x>)^2}
16	$
17	where $a$ is a field of scalars, and w a vector field.
18	Member fields and __init__ arguments : 
19	- a : an array of shape (n1,..,nk)
20	- w : an array of shape (vdim,n1,...,nk)
21	"""
22	def __init__(self,a,w):
23		a,w = (ad.asarray(e) for e in (a,w))
24		self.a,self.w =common_field((a,w),(0,1))
25
26	def norm(self,v):
27		v,a,w = common_field((ad.asarray(v),self.a,self.w),(1,0,1))
28		return np.sqrt(a**2*np.sum(v**2,axis=0) + np.sign(a)*np.maximum(lp.dot_VV(w,v),0.)**2)
29
30	def gradient(self,v):
31		v,a,w = common_field((ad.asarray(v),self.a,self.w),(1,0,1))
32		g = a**2*v + np.sign(a)*np.maximum(0.,lp.dot_VV(w,v))*w
33		return g/np.sqrt(lp.dot_VV(v,g))
34
35	def dual(self):
36		a2,s,w2 = self.a**2,np.sign(self.a),self._norm2w()
37		r2 = s*(1/a2-1/(a2+s*w2))/w2
38		return AsymIso(-1/self.a,np.sqrt(r2)*self.w)
39
40	@property
41	def vdim(self): return len(self.w)
42
43	@property
44	def shape(self): return self.a.shape
45
46	def _norm2w(self): return np.sum(self.w**2,axis=0)
47	def is_definite(self):
48		return np.where(self.a>0,1.,self.a**2-self._norm2w()) > 0
49	def anisotropy(self):
50		a2,w2 = self.a**2,self._norm2w()
51		return np.sqrt(np.where(self.a>0,1.+w2/a2,1./(1.-w2/a2)))
52	def cost_bound(self):
53		return np.where(self.a>0,np.sqrt(self.a**2+self._norm2w()),-self.a)
54
55	def rotate(self,r): return AsymIso(self.a,lp.dot_AV(r,self.w))
56	def with_cost(self,cost): return AsymIso(cost*self.a,cost*self.w)
57	def with_costs(self,costs):
58		if len(costs)>1 and not np.allclose(costs[1:],costs[0]):
59			raise ValueError("Costs must be identical along all axes for Metrics.Isotropic. Consider using Metrics.Diagonal class")
60		return self.with_cost(costs[0])
61
62	def model_HFM(self):
63		return 'AsymIso'+str(self.vdim)
64
65	@classmethod
66	def from_cast(cls,metric):
67		if isinstance(metric,cls): return metric
68		iso = Isotropic.from_cast(metric)
69		w = np.zeros_like(iso.cost,shape=(iso.vdim,)+iso.shape)
70		return cls(iso.cost,w)
71
72	def __iter__(self):
73		yield self.a
74		yield self.w
75
76	def make_proj_dual(self,**kwargs):
77		"""kwargs : passed to Riemann.make_proj_dual"""
78		vdim,a = self.vdim,self.a
79		proj_a = Isotropic(np.abs(a)).make_proj_dual()
80		eye = np.eye(vdim,like=a).reshape( (vdim,vdim)+(1,)*a.ndim )
81		proj_w=Riemann(a**2*eye+np.sign(a)*lp.outer_self(self.w)).make_proj_dual(**kwargs)
82
83		def proj(x):
84			x,w = common_field((x,self.w),depths=(1,1))
85			x_a = proj_a(x)
86			x_w = proj_w(x)
87			s = lp.dot_VV(w,x_a) > 0
88			return np.where(s[None],x_w,x_a)
89		return proj

A special case of AsymQuad metric taking the form $ F(x) = \sqrt{a^2 |x|^2+sign(a) max(0,)^2} $ where $a$ is a field of scalars, and w a vector field. Member fields and __init__ arguments :

a : an array of shape (n1,..,nk)
w : an array of shape (vdim,n1,...,nk)

AsymIso(a, w) View Source

22	def __init__(self,a,w):
23		a,w = (ad.asarray(e) for e in (a,w))
24		self.a,self.w =common_field((a,w),(0,1))

def norm(self, v): View Source

26	def norm(self,v):
27		v,a,w = common_field((ad.asarray(v),self.a,self.w),(1,0,1))
28		return np.sqrt(a**2*np.sum(v**2,axis=0) + np.sign(a)*np.maximum(lp.dot_VV(w,v),0.)**2)

Norm or quasi-norm defined by the class, often denoted $N$ in mathematical formulas. Unless incorrect data is provided, this member function obeys, for all vectors $u,v\in R^d$ and all $\alpha \geq 0$

$N(u+v) \leq N(u)+N(v)$
$N(\alpha u) = \alpha N(u)$
$N(u)\geq 0$ with equality iff $u=0$.

Broadcasting will occur depending on the shape of $v$ and of the class data.

def gradient(self, v): View Source

30	def gradient(self,v):
31		v,a,w = common_field((ad.asarray(v),self.a,self.w),(1,0,1))
32		g = a**2*v + np.sign(a)*np.maximum(0.,lp.dot_VV(w,v))*w
33		return g/np.sqrt(lp.dot_VV(v,g))

Gradient of the norm defined by the class.

def dual(self): View Source

35	def dual(self):
36		a2,s,w2 = self.a**2,np.sign(self.a),self._norm2w()
37		r2 = s*(1/a2-1/(a2+s*w2))/w2
38		return AsymIso(-1/self.a,np.sqrt(r2)*self.w)

Dual norm, mathematically defined by $N^*(x) = max\{ < x, y> ; N(y)\leq 1 \}$

vdim View Source

40	@property
41	def vdim(self): return len(self.w)

The ambient vector space dimension, often denoted $d$ in mathematical formulas.

shape View Source

43	@property
44	def shape(self): return self.a.shape

Dimensions of the underlying domain. Expected to be the empty tuple, or a tuple of length vdim.

def is_definite(self): View Source

47	def is_definite(self):
48		return np.where(self.a>0,1.,self.a**2-self._norm2w()) > 0

Attempts to check wether the data defines a mathematically valid norm.

def anisotropy(self): View Source

49	def anisotropy(self):
50		a2,w2 = self.a**2,self._norm2w()
51		return np.sqrt(np.where(self.a>0,1.+w2/a2,1./(1.-w2/a2)))

Anisotropy ratio of the norm denoted $N$. Defined as $$ \max_{|u| = |v| = 1} \frac {N(u)}{N(v)}. $$

def cost_bound(self): View Source

52	def cost_bound(self):
53		return np.where(self.a>0,np.sqrt(self.a**2+self._norm2w()),-self.a)

Upper bound on $N(u)$, for any unit vector $u$, where $N$ is the norm defined by the class.

def rotate(self, r): View Source

55	def rotate(self,r): return AsymIso(self.a,lp.dot_AV(r,self.w))

Rotation of the norm, by a given rotation matrix. The new unit ball is the direct image of the previous one.

def with_cost(self, cost): View Source

56	def with_cost(self,cost): return AsymIso(cost*self.a,cost*self.w)

Produces a norm $N'$ obeying $N'(x) = N(cost*x)$.

def with_costs(self, costs): View Source

57	def with_costs(self,costs):
58		if len(costs)>1 and not np.allclose(costs[1:],costs[0]):
59			raise ValueError("Costs must be identical along all axes for Metrics.Isotropic. Consider using Metrics.Diagonal class")
60		return self.with_cost(costs[0])

Produces a norm $N'$ defined by $$ N'(x) = N(costs * x) $$ where the multiplication is elementwise.

def model_HFM(self): View Source

62	def model_HFM(self):
63		return 'AsymIso'+str(self.vdim)

The name of the 'model' for parameter, as input to the HFM library.

@classmethod

def from_cast(cls, metric): View Source

65	@classmethod
66	def from_cast(cls,metric):
67		if isinstance(metric,cls): return metric
68		iso = Isotropic.from_cast(metric)
69		w = np.zeros_like(iso.cost,shape=(iso.vdim,)+iso.shape)
70		return cls(iso.cost,w)

Produces a metric by casting another metric of a compatible type.

def make_proj_dual(self, **kwargs): View Source

76	def make_proj_dual(self,**kwargs):
77		"""kwargs : passed to Riemann.make_proj_dual"""
78		vdim,a = self.vdim,self.a
79		proj_a = Isotropic(np.abs(a)).make_proj_dual()
80		eye = np.eye(vdim,like=a).reshape( (vdim,vdim)+(1,)*a.ndim )
81		proj_w=Riemann(a**2*eye+np.sign(a)*lp.outer_self(self.w)).make_proj_dual(**kwargs)
82
83		def proj(x):
84			x,w = common_field((x,self.w),depths=(1,1))
85			x_a = proj_a(x)
86			x_w = proj_w(x)
87			s = lp.dot_VV(w,x_a) > 0
88			return np.where(s[None],x_w,x_a)
89		return proj

kwargs : passed to Riemann.make_proj_dual

Inherited Members

agd.Metrics.base.Base: disassociate; anisotropy_bound; cos_asym; cos; angle; inv_transform; transform; rotate_by; with_speeds; with_speed; flatten; expand; to_HFM; from_HFM; from_generator; array_float_caster; norm2; gradient2; set_interpolation; at; make_proj; make_prox