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j-hartling
2026-06-18 14:50:08 +02:00
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main.tex
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@@ -1805,99 +1805,106 @@ adaptation. But the trade-off between intensity invariance and SNR likely goes
beyond the particular mechanisms along the pathway. After all, a transformation
is not expected to compress a range of different input intensities into a
constant output intensity without sacrificing some of the corresponding input
SNR. This suggests that the trade-off is a more general principle that applies
to any transformation that achieves or improves intensity invariance.
SNR. Accordingly, the trade-off likely is a more general principle that might
apply to any transformation that achieves or improves intensity invariance.
% Dependence of thresh-LP intensity invariance on threshold value (+unlimited SNR):
The second mechanism of intensity invariance consists of thresholding and
temporal averaging of $c_i(t)$ into $f_i(t)$. Here, the trade-off between
intensity invariance and SNR is mediated by the threshold value $\thr$. The
effects of $\thr$ on the intensity invariance and SNR of $f_i(t)$ are best
assessed if the mechanism is initially viewed in isolation. A lower
$\thr$~($\thr\to0$) improves the intensity invariance of $f_i(t)$ by shifting
the saturation point towards lower $\sca$. However, a lower $\thr$ also raises
the noise floor of $f_i(t)$ by including more of the pure-noise $c_i(t)$, which
decreases the SNR of $f_i(t)$. The distribution $\pci$ of the pure-noise
$c_i(t)$ is very close to a normal distribution with mean $\mu\approx0$ for all
kernels in the set. The value of the pure-noise $f_i(t)$ is hence 0.5 for
$\thr=0$ and decreases for higher $\thr$. If $\thr$ is set above the maximum of
$c_i(t)$, the pure-noise feature value is 0, which results in an "unlimited"
SNR of $f_i(t)$. In this case, any non-zero feature value that is sustained for
a sufficient duration could serve as indicator for the presence of $\soc(t)$ in
$\raw(t)$. Of course, this would require a fine evolutionary tuning of $\thr$
to the properties of the natural noise in a certain habitat to avoid false
positives.
assessed if the mechanism is viewed in isolation. A lower $\thr$~($\thr\to0$)
improves the intensity invariance of $f_i(t)$ by shifting the saturation point
towards lower $\sca$. The saturation level of $f_i(t)$ is mostly independent of
$\thr$, assuming that $\sca$ is sufficiently large. However, the lower $\thr$,
the more of the pure-noise $c_i(t)$ is included in $f_i(t)$ and hence the
higher the noise floor of $f_i(t)$, which decreases the SNR of $f_i(t)$. The
distribution $\pci$ of the pure-noise $c_i(t)$ is very close to a normal
distribution with mean $\mu\approx0$ for all kernels in the set. The value of
the pure-noise $f_i(t)$ is hence 0.5 for $\thr=0$ and decreases for higher
$\thr$. If $\thr$ is set above the maximum of $c_i(t)$, the pure-noise feature
value is 0, which results in an "unlimited" SNR of $f_i(t)$. In this case, any
non-zero feature value that is sustained for a sufficient duration could serve
as indicator for the presence of $\soc(t)$ in $\raw(t)$. Of course, this would
require a fine evolutionary tuning of $\thr$ to the properties of the natural
noise in a certain habitat in order to avoid false positives.
% Interaction between the two mechanisms of intensity invariance:
% Interaction between the two mechanisms of intensity invariance (expectations):
% (Also: Extremely important, but maybe too wordy?)
The combined effect of the two consecutive mechanisms of intensity invariance
depends on which mechanism results in a lower saturation point. In case of
$f_i(t)$, it is necessary to distinguish between its intrinsic saturation
point~(the saturation point that the second mechanism can achieve in isolation)
and its actual saturation point~(including the effects of the first mechanism).
The same distinction applies to the saturation level of $f_i(t)$. The intrinsic
saturation point of $f_i(t)$ increases with increasing $\thr$. The intrinsic
saturation level of $f_i(t)$ is largely independent of $\thr$, assuming that
$\thr$ is sufficiently small or $\sca$ is sufficiently large. If the intrinsic
The intensity invariance of $f_i(t)$ is not only determined by the second
mechanism but by the interaction between the two consecutive mechanisms along
the pathway. This interaction is difficult to assess systematically due to the
multitude of involved parameters. A basic expectation is that the combined
effects of the two mechanisms mostly depend on which mechanism achieves a lower
saturation point, assuming that $f_i(t)$ is always intensity-invariant if
$\adapt(t)$ is already intensity-invariant. Furthermore, it is necessary to
distinguish between the intrinsic saturation point of $f_i(t)$ --- the
saturation point that the second mechanism can achieve in isolation --- and its
actual saturation point including the effects of the first mechanism. The same
distinction applies to the saturation level of $f_i(t)$. If the intrinsic
saturation point of $f_i(t)$ is lower than the saturation point of $\adapt(t)$,
the second mechanism will take precedence over the first mechanism. In this
case, $f_i(t)$ will reach the intrinsic saturation level at the intrinsic
$f_i(t)$ is expected to reach the intrinsic saturation level at the intrinsic
saturation point. In contrast, if the intrinsic saturation point of $f_i(t)$ is
higher than the saturation point of $\adapt(t)$, the first mechanism will take
precedence over the second mechanism. In this case, the actual saturation point
of $f_i(t)$ will be determined by the saturation point of $\adapt(t)$ rather
than the intrinsic saturation point of $f_i(t)$. This has no detrimental effect
on the intensity invariance of $f_i(t)$. However, a lower saturation point of
$f_i(t)$ means that the actual saturation level of $f_i(t)$ will be lower than
its intrinsic saturation level. Moreover, the saturation level of $f_i(t)$ will
not be independent of $\thr$ anymore but will decrease with increasing $\thr$.
A lower saturation level of $f_i(t)$ does not necessarily impair the SNR of
$f_i(t)$ --- $f_i(t)$ can still achieve an arbitrarily high SNR by setting
$\thr$ just above the maximum pure-noise $c_i(t)$. However, a lower saturation
level of $f_i(t)$ does mean that the range of possible feature values that
$f_i(t)$ can take on is restricted compared to the case where $f_i(t)$ can
reach its intrinsic saturation level. In summary, the interaction between the
two mechanisms of intensity invariance along the pathway can have unfavorable
consequences for the overall system if the first mechanism takes precedence
over the second mechanism. However, this interaction does not so much affect
the intensity invariance or the SNR of $f_i(t)$ but rather constraints the part
of the feature space that is available for species-specific song
representation.
higher than the saturation point of $\adapt(t)$, $f_i(t)$ is expected to
saturate at the lower saturation point of $\adapt(t)$ instead. This has no
detrimental effect on the intensity invariance of $f_i(t)$. However, $f_i(t)$
is then also expected to saturate below its intrinsic saturation level.
Moreover, the saturation level of $f_i(t)$ is not independent of $\thr$ anymore
but decreases with increasing $\thr$. A lower saturation level of $f_i(t)$ is
not necessarily detrimental to the SNR of $f_i(t)$ --- $f_i(t)$ can still
achieve an arbitrarily high SNR by setting $\thr$ just above the maximum
pure-noise $c_i(t)$. More importantly, a lower saturation level of $f_i(t)$
also means that the range of possible feature values that $f_i(t)$ can take on
is limited compared to the case where $f_i(t)$ can reach its intrinsic
saturation level. This effectively restricts the part of the feature space that
is available for species-specific song representation. The interaction between
the two mechanisms of intensity invariance could therefore have unfavorable
consequences if the first mechanism results in a lower saturation point than
the second mechanism.
% Check log-axis histogram counts!
% Why do so many features have a lower saturation point than adapt if so many
% do not reach the intrinsic saturation level??
Judging from the distribution of saturation points across the set of $f_i(t)$,
both interactions between the two mechanisms appear to be present in the
current pathway. A number of $f_i(t)$ achieve a lower saturation point than
$\adapt(t)$, which indicates that the second mechanism takes precedence over
the first mechanism. These cases raise the question whether the first mechanism
is actually necessary for the overall system if the second mechanism can
apparently achieve intensity invariance with a lower saturation point. There
are also some $f_i(t)$ whose saturation point matches the saturation point of
$\adapt(t)$, which indicates that the first mechanism takes precedence over the
second mechanism. These cases raise the question whether intensity invariance
by the first mechanism --- while achieving a lower saturation point than the
second mechanism --- is actually beneficial
% Interaction between the two mechanisms of intensity invariance (current results):
The saturation point and saturation level of a feature in the set varies with
the specific kernel.
The combined effects of the two mechanisms on the intensity invariance of a
specific feature in the set vary between different kernels
Based on the current results, it is difficult to assess which of the two
mechanisms has a stronger effect on the intensity invariance of a specific
feature in the set.
The combined effects of the two mechanisms on the intensity invariance of a
specific feature in the set vary between different kernels. It is difficult to
assess which of the two mechanisms achieves a lower saturation point for a
specific feature. On the one hand, the distribution of saturation levels across
the feature set is not symmetric around a feature value of 0.5, which is the
case if the logarithmic compression along the pathway is disabled. This result
indicates that a number of features does not reach the intrinsic saturation
level, which suggests that the intensity invariance of these features is
determined by the first mechanism rather than the second mechanism. One the
other hand, the distribution of saturation points across the feature set
indicates that a number of features does indeed achieve a lower saturation
point than the preceeding representations. This result suggests that the
intensity invariance of these features is determined by the second mechanism
rather than the first mechanism. In either case, the question arises to what
extent two consecutive mechanisms of intensity invariance are actually
beneficial for the overall system.
These cases raise the question whether the first mechanism is actually
necessary for the overall system if the second mechanism can apparently achieve
intensity invariance with a lower saturation point. These cases raise the
question whether intensity invariance by the first mechanism --- while
achieving a lower saturation point than the second mechanism --- is actually
beneficial
The saturation point of $f_i(t)$ varies between different kernels in the set. A
number of $f_i(t)$ achieve a lower saturation point than $c_i(t)$ --- and hence a lower saturation point than $\adapt(t)$ --- which
indicates that the second mechanism takes precedence over the first mechanism.
Some $f_i(t)$ exhibit similar or only marginally lower saturation points than
The saturation points of $f_i(t)$ across the set are distributed over a much
wider range than those of the preceeding $c_i(t)$,
which suggests that the
interaction between the two mechanisms is specific to individual kernels. A
number of $f_i(t)$ achieve a lower saturation point than the respective
$c_i(t)$,
whereas some $f_i(t)$ exhibit similar or only marginally lower
saturation points.
In these cases, the question arises to what extent two
consecutive mechanisms of intensity invariance are actually beneficial for the
overall system.

13
python/fig_invariance_full.py
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@@ -442,6 +442,7 @@ for stage in stages:
# Indicate saturation point(s):
if stage in ['log', 'inv', 'conv', 'feat']:
# Get and plot single curve saturation point:
ind = get_saturation(curve, **plateau_settings)[1]
crit_inds[stage] = ind
scale = scales[ind]
@@ -452,6 +453,13 @@ for stage in stages:
transform=raw_axes[0].get_xaxis_transform())
raw_axes[0].vlines(scale, raw_axes[0].get_ylim()[0], curve[ind],
color=color, **plateau_line_kwargs)
if stage in ['conv', 'feat']:
# Get and log distribution of swarm saturation points:
inds = np.array(get_saturation(measure, **plateau_settings)[1])
if np.isnan(inds).sum():
print('WARNING: Found NaN saturation point(s)!')
inds = inds[~np.isnan(inds)].astype(int)
crit_scales_swarm[stage] = scales[inds]
## NORMALIZED MEASURE:
@@ -476,11 +484,6 @@ for stage in stages:
fill_kwargs = dist_fill_kwargs | dict(color=color)
y_dist(base_insets[i1], measure[-1], nbins=100, log=True,
line_kwargs=line_kwargs, fill_kwargs=fill_kwargs)
# Get and log distribution of saturation points:
inds = np.array(get_saturation(measure, **plateau_settings)[1])
if np.isnan(inds).sum():
inds = inds[~np.isnan(inds)].astype(int)
crit_scales_swarm[stage] = scales[inds]
if stage == 'feat':
# Plot distribution of saturation points on shared bins:
bin_lims = [0.01, 1.1 * max([s.max() for s in crit_scales_swarm.values()])]