-
Notifications
You must be signed in to change notification settings - Fork 0
/
experiment4_hypercontractivity_bounds.m
145 lines (136 loc) · 4.42 KB
/
experiment4_hypercontractivity_bounds.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
%% Visualize bounds in Theorem 11
%% Paper: "Estimation Contracts for Outlier-Robust Geometric Perception"
%% Luca Carlone, Nov 7, 2022
%% plot coefficients C1 and C2:
close all
clc
N = 50;
Ct_pow_t_set = [1 2 4 6];
k = 4;
for j=1:length(Ct_pow_t_set)
t = k/2;
Ct_pow_t = Ct_pow_t_set(j); % Ct^t;
betaMax = ( 1/(Ct_pow_t*2^(3*k-1)) )^(1 / (k/2 - 1)) - 1e-6 % minus some numerical margin
beta = linspace(0,betaMax,100000);
C1 = zeros(length(beta),1);
C2 = zeros(length(beta),1);
for i=1:length(beta)
betai = beta(i);
C1(i) = C1_pow_2_over_k(betai, Ct_pow_t, k);
C2(i) = C2_pow_2_over_k(betai, Ct_pow_t, k);
end
%% plot C1
figure(1)
semilogy(beta,C1,'-r','linewidth',2)
hold on
grid on
ylimits = [2*10^-3 40];
xlabel('Outlier rate $\beta$','interpreter','latex')
ylabel('$C_1(4,\beta)^{1/2}$','interpreter','latex')
plot([betaMax betaMax],ylimits,'--k','linewidth',1)
s = sprintf(' C(t)^t = %d ',Ct_pow_t);
switch Ct_pow_t
case 1
text(beta(end),C1(end)+6,s,'HorizontalAlignment','right')
case 2
text(beta(end),C1(end)+4,s,'HorizontalAlignment','left')
case 4
text(beta(end),C1(end)+2,s,'HorizontalAlignment','left')
case 6
text(beta(end),C1(end)+2,s,'HorizontalAlignment','right')
end
ylim(ylimits)
xlim([-1e-5 5*1e-4])
%% plot C2
figure(2)
semilogy(beta,C2,'-b','linewidth',2)
hold on
grid on
ylimits = [1*10^-3 20];
xlabel('Outlier rate $\beta$','interpreter','latex')
ylabel('$C_2(4,\beta)^{1/2}$','interpreter','latex')
plot([betaMax betaMax],ylimits,'--k','linewidth',1)
ylim([10^-5 10^4])
s = sprintf(' C(t)^t = %d ',Ct_pow_t);
switch Ct_pow_t
case 1
text(beta(end),C2(end)+1,s,'HorizontalAlignment','right')
case 2
text(beta(end),C2(end)+1,s,'HorizontalAlignment','left')
case 4
text(beta(end),C2(end)+1,s,'HorizontalAlignment','left')
case 6
text(beta(end),C2(end)+1,s,'HorizontalAlignment','right')
end
ylim(ylimits)
xlim([-1e-5 5*1e-4])
end
%% test for increasing k
Ct_pow_t = 6; % Ct^t;
for k=[4 6 8 10]
t = k/2;
betaMax = ( 1/(Ct_pow_t*2^(3*k-1)) )^(1 / (k/2 - 1)) - 1e-6 % minus some numerical margin
beta = linspace(0,betaMax,100000);
C1 = zeros(length(beta),1);
C2 = zeros(length(beta),1);
for i=1:length(beta)
betai = beta(i);
C1(i) = C1_pow_2_over_k(betai, Ct_pow_t, k);
C2(i) = C2_pow_2_over_k(betai, Ct_pow_t, k);
end
%% plot C1
figure(3)
semilogy(beta,C1,'-r','linewidth',2)
ylimits = [7*1e-5 30];
hold on
grid on
xlabel('Outlier rate $\beta$','interpreter','latex')
ylabel('$C_1(k,\beta)^{2/k}$','interpreter','latex')
plot([betaMax betaMax],ylimits,'--k','linewidth',1)
s = sprintf(' k = %d ', k);
switch k
case 4
text(beta(end),C1(end)+5,s,'HorizontalAlignment','left')
case 6
text(beta(end),C1(end)+5,s,'HorizontalAlignment','left')
case 8
text(beta(end),C1(end)+5,s,'HorizontalAlignment','left')
case 10
text(beta(end),C1(end)+5,s,'HorizontalAlignment','right')
end
ylim(ylimits)
xlim([-1e-4 4.5*1e-3])
%% plot C2
figure(4)
semilogy(beta,C2,'-b','linewidth',2)
ylimits = [3*1e-5 10];
hold on
grid on
xlabel('Outlier rate $\beta$','interpreter','latex')
ylabel('$C_2(k,\beta)^{2/k}$','interpreter','latex')
plot([betaMax betaMax],ylimits,'--k','linewidth',1)
ylim([10^-5 10^4])
s = sprintf(' k = %d ',k);
switch k
case 4
text(beta(end),C2(end)+1,s,'HorizontalAlignment','left')
case 6
text(beta(end),C2(end)+1,s,'HorizontalAlignment','left')
case 8
text(beta(end),C2(end)+1,s,'HorizontalAlignment','left')
case 10
text(beta(end),C2(end)+1,s,'HorizontalAlignment','right')
end
ylim(ylimits)
xlim([-1e-4 4.5*1e-3])
end
function value = C1_pow_2_over_k(betai, Ct_pow_t, k)
value = (betai^(k/2-1) * Ct_pow_t * 2^( 3*k-1 ) ) / ...
( 1 - betai^(k/2-1) * Ct_pow_t * 2^( 3*k-1 ) );
value = value^(2/k);
end
function value = C2_pow_2_over_k(betai, Ct_pow_t, k)
value = ((2*betai)^(k/2-1) * ( 2^k + Ct_pow_t * 2^( 2*k ) ) ) / ...
( 1 - betai^(k/2-1) * Ct_pow_t * 2^( 3*k-1 ) );
value = value^(2/k);
end