Projected distances for multi-parameter persistence modules

Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℝ</mml:mi> </mml:math> . Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> -linear ISM and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> -linear SCD which are projected distances tailored for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> -sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous since our approach does not require computing nor storing an entire <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -persistence module.