Updated derivation with improved numerical stability

[seanlaw]

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[gitnotebooks]

Found 1 changed notebook. Review the changes at https://app.gitnotebooks.com/stumpy-dev/stumpy/pull/1180

[seanlaw]

@NimaSarajpoor Would you mind taking a look when you have some time? Thanks in advance!

You may need to download the file in order to get the LaTex to render properly

[NimaSarajpoor]

@seanlaw
It was great to see all the steps involved in deriving equation (91). Thanks for putting them together so clearly! I have shared a few comments for your consideration.

[NimaSarajpoor]

It should be worth it to better demonstrate/explain the benefit of achieving this new equation. We can show that the new equation (91) has LARGER UPDATE and therefore should be more stable as it has less catastrophic cancelation.

Let's start with the equation (40), i.e.

$$cov(Q_{i,m}, T_{i,m}) = cov(Q_{i-1,m}, T_{i-1,m}) + update$$

Let's multiply both sides by m:

$$ m * cov(Q_{i,m}, T_{i,m}) = m * cov(Q_{i-1,m}, T_{i-1,m}) + m * update $$

Recall that m * cov is basically the summation part of "centered sum-of-products". So, this means the update term in the equation (91), i.e. $df^{T}.dg^{Q} + df^{Q}.dg^{T}$, is equivalent to m * update (NOT update). This quickly shows that the new equation has LARGER UPDATE.

[NimaSarajpoor]

Let's see what else we can get from the relationship between equation (40) and (91). Let's start from equation (40) again:

$$cov(Q_{i,m}, T_{i,m}) = cov(Q_{i-1,m}, T_{i-1,m}) + \frac{m-1}{m^{2}}U$$

where

$$ U = [(q_{i+m-1} - \mu_{Q_{i,m-1}})(t_{i+m-1} - M_{T_{i,m-1}}) - (q_{i-1} - \mu_{Q_{i,m-1}}) (t_{i-1} - M_{T_{i,m-1}})]$$

---

Let's multiply both sides of equation (40) by $m$:

$$m * cov(Q_{i,m}, T_{i,m}) = m * cov(Q_{i-1,m}, T_{i-1,m}) + \frac{m-1}{m}U$$

Again, recall that m * cov is basically the summation part of "centered sum-of-products". So, if I compare the equation above with the equation (91), I can see:

$$ \frac{m-1}{m}U = df^{T}.dg^{Q} + df^{Q}.dg^{T} $$

Note that $\frac{m-1}{m} < 1$. If $U$ is small, then $\frac{m-1}{m}U$ is smaller, meaning the value on the right hand side of the equation above is smaller than $U$. Therefore, if we say that $U$ involves the subtraction of two nearby numbers and that can lead to catastrophic cancellation, then, in that scenario, we will have "catastrophic cancellation" on the right hand side too as that is even smaller!

Can we simply use the update term $\frac{m-1}{m}U$ when we use the summation part of "centered sum-of-products"?

[seanlaw]

Maybe it's an issue on my end when things are being rendered in my Jupyter notebook but I'm only seeing a single equation number for a block of related/multiline equations:

If you are seeing the same thing, maybe we can refer to the lines as (14.1) - (14.5) (to mean lines 1-5 in equation block 14)? Otherwise, I am counting equations one-by-one each time :(

Or maybe you have a different workflow?

[seanlaw]

Can we simply use the update term $\frac{m−1}{m}U$ when we use the summation part of "centered sum-of-products"?

I've tried reading your comments a few times but I am unable to get your point.

Therefore, if we say that $U$ involves the subtraction of two nearby numbers and that can lead to catastrophic cancellation, then, in that scenario, we will have "catastrophic cancellation" on the right hand side too as that is even smaller!

I feel like I'm only reading this as a statement of fact but I am getting the sense that you have a much stronger point/argument that I am overlooking :(

You started with:

It should be worth it to better demonstrate/explain the benefit of achieving this new equation.

Maybe I've been staring at the equations too long but I'm not able to see "how" this is demonstrating the benefit any further (i.e., what is the special insight that "proves" the benefit clearly). I know you so I can feel that there is something (intuition?) there. I'm just not smart enough to infer your intuition but I would like to hear more.

[NimaSarajpoor]

Or maybe you have a different workflow?

I didn't download the notebook. I just opened the file on Github from your branch. It renders the file differently and I didn't know that it affects the equations numbers! Apologies for the inconvenience.

Please allow me to download the updated notebook and read it again to see the added information. I will use the equation numbers from there.

[NimaSarajpoor]

I downloaded and read the (updated) notebook. I opened the file in notebook this time and cannot see equation numbers. Strange!

---

I've tried reading your comments a few times but I am unable to get your point.

If I understand correctly, the gist of this PR is that:

• The correction term $[(q_{i+m-1} - \mu_{Q_{i,m-1}})(t_{i+m-1} - M_{T_{i,m-1}}) - (q_{i-1} - \mu_{Q_{i,m-1}}) (t_{i-1} - M_{T_{i,m-1}})]$, which involves the subtraction of two nearby numbers, is small and can lead to catastrophic cancellation.
• The proposal is to use $df^{T}.dg^{Q} + df^{Q}.dg^{T}$ instead (in a new recurrence relation).

HOWEVER, it can be shown that $df^{T}.dg^{Q} + df^{Q}.dg^{T}$ is actually smaller than the original correction term $[(q_{i+m-1} - \mu_{Q_{i,m-1}})(t_{i+m-1} - M_{T_{i,m-1}}) - (q_{i-1} - \mu_{Q_{i,m-1}}) (t_{i-1} - M_{T_{i,m-1}})]$. IMO, it means that if the original "correction" term $[(q_{i+m-1} - \mu_{Q_{i,m-1}})(t_{i+m-1} - M_{T_{i,m-1}}) - (q_{i-1} - \mu_{Q_{i,m-1}}) (t_{i-1} - M_{T_{i,m-1}})]$ is small and can lead to catastrophic cancelation, then $df^{T}.dg^{Q} + df^{Q}.dg^{T}$ is smaller, meaning $(df^{T}.dg^{Q}) - (-df^{Q}.dg^{T})$ can also lead to catastrophic cancelation and it can be even worse!

So, we might be better off using the original correction term when updating the $\overline{QT}$.