![](bg9.png)
ance of the model to a variety of pitch phenomena to
the model.
However, the given implementation of the model
hampers its ability to robustly capture this distinct re-
lationship between the low-frequency and the high-
frequency units with higher precision because of two
reasons. First, the restricted number of sparse com-
ponents inherently introduces a trade-off in their fre-
quency resolution as well as in their capacity to en-
code the phase of the signal. Second, the model
training implements a block based approach to signal
encoding, with the sound segments being randomly
drawn from the set of available sentences, irrespec-
tive of the phase structure of the signals. Therefore,
the representation of the higher-order structure related
to temporal pitch is highly sensitive to the learning
process. We plan on addressing these issues in future
work.
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APPENDIX
Inference in the Fully Extended Density
Component Model
As described previously, the alterations of the origi-
nal density component model affect the encoding and
learning procedures, while the generative model itself
remains the same. As the transformation from the
sound to the higher-order representation vdc is fun-
damentally nonlinear, the optimal coefficient values
for the representation cannot be expressed in closed
form
2
. In order to encode a given signal, the MAP es-
timation of the sparse (SC) and the density component
coefficients (DC) is illustrated in figure 6:
1. Choose a whitened sound extract x
w
of length T .
2. Generate the corresponding spectrogram S
x
of
temporal length T
S
> T and frequency resolution
F
res
, using a logarithmic scaling of the frequen-
cies.
3. Calculate the gist information:
(a) Project S
x
into the space of the first 50 principal
components explaining approximately 95% of
the variance:
u
G
= W
S
pca
S
x
, W
S
pca
= D
−1/2
S
E
T
S
, (12)
2
Closed form means that the expression can be written
analytically in terms of a bounded number of certain well-
known functions (i.e. no infinite series, etc.)
PITCH-SENSITIVE COMPONENTS EMERGE FROM HIERARCHICAL SPARSE CODING OF NATURAL SOUNDS
227