Geometric bounds for persistence

In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Y"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Y}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we analyze the topological features arising in the family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript bullet Baseline left-parenthesis script upper X subset-of bold upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mo> ∙ </mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Y</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {N}_\bullet (\mathcal {X}\subset \mathbf {Y})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of nested neighborhoods of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Y"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Y}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and provide several geometric bounds on their persistence (lifespans). We begin by examining the lifespans of these homology classes in terms of their filling radii in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Y"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {Y}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\ell ^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -principal components of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> : the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Čech and Vietoris–Rips extinction times of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by relating <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to its convex hull and to its tight span, respectively.