From 2ea59c520bc84caf91ef34a01a7e47918af83924 Mon Sep 17 00:00:00 2001 From: j-hartling Date: Wed, 17 Jun 2026 18:20:04 +0200 Subject: [PATCH] Writing discussion. --- main.tex | 113 ++++++++++++++++++++++++++++++++++++++----------------- 1 file changed, 78 insertions(+), 35 deletions(-) diff --git a/main.tex b/main.tex index c08c3b4..aacdf52 100644 --- a/main.tex +++ b/main.tex @@ -1781,56 +1781,99 @@ 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. +% 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 $\thr$. A lower +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 $k_i(t)$. The value of the pure-noise $f_i(t)$ is hence 0.5 for +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 -$\raw(t)$, although at the cost of a higher saturation point. 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. +$\raw(t)$. 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. -The saturation level of $f_i(t)$ is independent of $\thr$ as long as the -intensity invariance by the previous mechanism is neglected. +% Interaction between the two mechanisms of intensity invariance: +% (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 +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 +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. + +% 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 + +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 -If $f_i(t)$ can achieve an arbitrarily high SNR, it can counteract the -comparably low SNR of $\adapt(t)$ - -The maximum of the pure-noise $c_i(t)$ is assumed to be very -small due to the various SNR improvements along the pathway, so that the -required increase in $\thr$ and hence the saturation point of $f_i(t)$ is not -expected to be substantial. - - -% \newpage -% \subsection{Intensity invariance versus intensity invariance} - -Two consecutive mechanisms of intensity invariance do not necessarily add up to -a stronger overall intensity invariance. If the first mechanism results in a -lower saturation point than the second mechanism by itself, the saturation -point of feature $f_i(t)$ will be determined solely by the first mechanism. In -this case, the saturation level of $f_i(t)$ will conform to the intensity that -$f_i(t)$ can reach for the given saturation point rather than the intrinsic -saturation level of $f_i(t)$. Conversely, if the second mechanism results in a -lower saturation point than the first mechanism, both the saturation point and -the saturation level of $f_i(t)$ will be determined by the second mechanism. The saturation points of $f_i(t)$ across the set are distributed over a much -wider range than those of the preceeding kernel responses $c_i(t)$, 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. +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. From a computational perspective, the answer could be that logarithmic compression and adaptation is a necessary preprocessing step towards robust