Consider a W shaped function with local minimums at $y=1$ and $y=2$ and local maximum at $y=3$. When we look at the persistence diagram induced by the lower level sets of this function,
- Two topological features are born at $y=1$ and $y=2$
- One of them dies at $y=3$ and the other never dies.
So, is it true that two possible persistence diagrams exist?
- Diagram$1$: $[1,\infty) , \quad [2,3)$
- Diagram$2$: $[1,3) , \quad [2, \infty)$
This becomes very problematic when we say "features with lifetime $\leq 1.5$ are topological noise." because in one case we have one significant component, whereas in the other case there are two significant connected components.
To add a bit more notation, I'll assume you have a piece-wise linear function $f \colon [0,4]\to\mathbb{R}$ with $f(0)=4$, $f(1)=1$, $f(2)=3$, $f(3)=2$, and $f(4)=4$, and $f$ is linear everywhere in-between these values. The graph of $f$ is this "W" shape that you mention, with local minima of heights 1 and 2, and with an interior local maximum of height 3. The sublevelset persistent homology of this function $f$ has two bars, $[1,\infty)$ and $[2,3)$. This is your Diagram 1. Your Diagram 2 is not the correct sublevelset persistent homology of this function $f$. This follows (for example) from the Elder Rule in Chapter VII of "Computational Topology: An Introduction" by Edlesbrunner and Harer; see for example https://books.google.com/books?id=MDXa6gFRZuIC&pg=PA151&lpg=PA151&dq=persistent+homology+elder+rule&source=bl&ots=CRYq86THmC&sig=ACfU3U2Ew2cUCPcMiG45kLefycNntJHXTA&hl=en&sa=X&ved=2ahUKEwi4qdfYy6PqAhWtAp0JHaKcAKIQ6AEwE3oECAsQAQ#v=onepage&q=elder%20rule&f=false. One can also see that these are the correct bars by running the persistence algorithm by hand. More generally, the decomposition theorem in persistent homology says that the correct persistent homology bars are always uniquely defined.