Database Reference
In-Depth Information
Lemma 5.5
Assuming the
w
i
,
1
≤
i
≤
k
are orthonormal, we have
E
t
=
t
τ
=1
E
t−
1
+
t
2
=
t−
1
y
τ
y
t
.
t
Proof.
If the
w
i
,
1
≤
i
≤
k
are orthonormal, then it follows easily that
2
=
2
=
y
τ,
1
w
1
2
+
+
y
τ,k
w
k
2
=
x
τ
y
τ,
1
w
1
+
···
+
y
τ,k
w
k
···
y
τ,
1
+
2
(Pythagorean theorem and normality). The
result follows by summing over
τ
.
+
y
τ,k
=
···
y
τ
It can be shown that algorithm TrackW maintains orthonormality
without the need for any extra steps (otherwise, a simple re-orthonormali-
sation step at the end would su
ce).
From the user's perspective, we have a low-energy and a high-energy
threshold,
f
E
and
F
E
, respectively. We keep enough hidden variables
k
,
so the retained energy is within the range [
f
E
·
E
t
]. Whenever
we get outside these bounds, we increase or decrease
k
. In more detail,
the steps are:
E
t
,
F
E
·
1 Estimate the full energy
E
t
+1
, incrementally, from the sum of
squares of
x
τ,i
.
2 Estimate the energy
E
(
k
)
of the
k
hidden variables.
3 Possibly, adjust
k
. We introduce a new hidden variable (update
k
k
+1) if the current hidden variables maintain too little energy,
i.e.,
E
(
k
)
<f
E
E
. We drop a hidden variable (update
k
←
←
k
−
1),
if the maintained energy is too high, i.e.,
E
(
k
)
>F
E
E
.
The energy thresholds
f
E
and
F
E
arechosenaccordingtorecommenda-
tions in the literature [33, 20]. We use a lower energy threshold
f
E
=0
.
95
and an upper threshold
F
E
=0
.
98. Thus, the reconstruction
x
t
retains
between 95% and 98% of the energy of
x
t
.
The following lemma proves that the above algorithm guarantees the
relative reconstruction error is within the specified interval [
f
E
,F
E
].
Lemma 5.6
The relative squared error of the reconstruction satisfies
τ
=1
2
x
τ
−
x
τ
t
1
−
F
E
≤
≤
1
−
f
E
.
x
τ
2
2
=
Proof.
From the orthogonality of
x
τ
and
x
τ
−
x
τ
we have
x
τ
−
x
τ
2
(by Lemma 5.5). The result follows by
summing over
τ
and from the definitions of
E
and
E
.
2
2
=
2
x
τ
−
x
τ
x
τ
−
y
τ
