Vectorization2 #378

ammedmar · 2020-03-23T15:17:42Z

Reference issues/PRs
Fixes #361

Signed-off-by: ammedmar <anibal@medina-mardones.com>

Signed-off-by: ammedmar <ammedmar@gmail.com>

CLAassistant · 2020-03-23T15:17:48Z

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ulupo · 2020-03-23T15:24:21Z

doc/theory/glossary.tex

@@ -65,7 +62,7 @@

 	A set $\{v_0, \dots, v_n\} \subset \mathbb{R}^N$ is said to be \textit{geometrically independent} if the vectors $\{v_0-v_1, \dots, v_0-v_n\}$ are linearly independent. In this case, we refer to their convex closure as a \textit{simplex}, explicitly
 	\begin{equation*}
-	\lbrack v_0, \ldots , v_n \rbrack = \left\{ \sum c_i (v_0 - v_i)\ \big|\ c_1+\dots+c_n = 1,\ c_i \geq 0 \right\}
+	[v_0, \dots , v_n] = \left\{ \sum c_i (v_0 - v_i)\ \big|\ c_1+\dots+c_n = 1,\ c_i \geq 0 \right\}


\lbrac and \rbrac are necessary unfortunately...

Oh, ok. I'll change it everywhere then.

A small correction.
No need to do it everywhere: I introduced the change only here, because it was not properly parsed by pandoc: it seems like it has a problem with [ at the beginning of the equation. For example, d_i[...]... is ok.

ulupo · 2020-03-23T15:26:51Z

doc/theory/glossary.tex

+
+	\subsection*{$L^p$-norm:} \label{l^p_norm}	
+
+	...


What's the plan for this subsection? To leave it to another PR?

It depends on the persistence image entry, TBW concomitantly.

I should mention that I included it since it is referenced in persistence landscape.

Does the reference still exist?
I am not questioning the need for this section.

Yes, the word "p-integrable" should link to L^p

Signed-off-by: ammedmar <ammedmar@gmail.com>

wreise

Thanks!
A few comments about labels.

What about L^p - is it still needed? can i take it?

wreise · 2020-03-26T17:23:22Z

doc/theory/glossary.tex

+	\end{equation*}	
+	and $c_+ := \max(c,0)$. The function $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}.	
+
+	Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by $1/\sqrt{2}$.	


Am i correct to say that you are referring to $\Lambda$ and not $\lambda$ ?

After reviewing, I think I can do better than I did a few months go explaining this. Will give it another go.

OK, sorry this took me a few days. It should ready for review.

wreise · 2020-03-26T17:29:51Z

doc/theory/glossary.tex

+	vectorization of the set of persistence diagrams with target the vector space of real-valued function on $\mathbb N \times \mathbb R$. For any $p = 1,\dots,\infty$ we can restrict attention to persistence diagrams $D$ whose associated persistence landscape $\lambda$ is
+	%\hyperref[l^p norm]{$p$-integrable}
+	$p$-integrable, that is to say,	
+	\begin{equation} \label{equation: persistence landscape norm}	


Would mind if we called it equation:persistence_landscape_norm instead?

wreise · 2020-03-26T17:32:47Z

doc/theory/glossary.tex

+	\begin{equation*}	
+	||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p}	
+	\end{equation*}	
+	is finite. In this case we refer to (\ref{equation: persistence landscape norm}) as the


would \eqref{equation: persistence landscape norm} work for you?

Sure. Should I replace all references to equations by "\eqref"?

wreise · 2020-03-26T17:34:08Z

doc/theory/glossary.tex

@@ -396,54 +387,66 @@

 	The name is inspired from the case when the persistence diagram comes from persistent homology.

-	\subsection*{Distances, inner products and kernels} \label{metric_inner_product_and_kernel}
+	\subsection*{Metric space} \label{metric space}	


As above, would you mind metric_space ?

I will change it. Sorry for the slip.

wreise · 2020-03-26T17:36:32Z

doc/theory/glossary.tex

+	\begin{equation*}	
+	d : X \times X \to \mathbb R	
+	\end{equation*}	
+	is said to be a \textit{metric space} if the values of $d$ are all non-negative and for all $x,y,z \in X$ ans $a \in \mathbb R$	


Did you mean and?
Also, i can't find where a is used.

Good catch!

wreise · 2020-03-26T17:40:13Z

doc/theory/glossary.tex

-	In this case the function $\langle -, - \rangle$ is referred to as the \textit{inner product} and the function given by
-	\begin{equation*}
-	||u|| = \sqrt{\langle u, u \rangle}
+	is said to be an \textit{inner product space} if for all $u,v,w \in V$


Or, a\in\mathbb R was meant to be here?

Yep, will get it fixed.

wreise · 2020-03-26T17:43:08Z

doc/theory/glossary.tex

+
+	\subsection*{$L^p$-norm:} \label{l^p_norm}	
+
+	...


Does the reference still exist?
I am not questioning the need for this section.

Signed-off-by: ammedmar <ammedmar@gmail.com>

wreise · 2020-04-03T07:31:26Z

doc/theory/glossary.tex


-	Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by $1/\sqrt{2}$.	
+	We describe the graph of each $\lambda_k$ intuitively. For each $i \in I$, draw an isosceles triangle with base the interval $(b_i, d_i)$ on the horizontal $t$-axis, and sides with  slope  1  and $-1$.  This  subdivides  the  plane  into  a  number  of  polygonal regions.  Label each of these regions by the number of triangles containing it.  If $P_k$ is the union of the polygonal regions with values at least $k$, then the graph of $\lambda_k$ is the upper contour of $P_k$, with $\lambda_k(a) = 0$ if the vertical line $t=a$ does not intersect $P_k$.


Reads well!
There is a number of double-spaces here, but latex still renders it properly as since spaces, so it's ok imo.

Thanks, now fixed

wreise · 2020-04-03T07:33:03Z

doc/theory/glossary.tex

 	||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p}	
 	\end{equation}	
 	where
 	\begin{equation*}	
 	||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p}	
 	\end{equation*}	
-	is finite. In this case we refer to (\ref{equation: persistence landscape norm}) as the
+	is finite. In this case we refer to (\eqref{equation:persistence_landscape_norm}) as the


I would remove the parentheses, because eqref already produces them. In this case, in the pdf, you see ((1)).

Signed-off-by: ammedmar <ammedmar@gmail.com>

wreise

LGTM, maybe @ulupo would like to review before you merge.

ulupo

Just asking for some minor changes for future book-keeping, but looks great otherwise!

ulupo · 2020-04-05T19:31:12Z

doc/theory/glossary.tex

@@ -307,76 +304,69 @@
 	\begin{equation*}
 	\sup_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_{\infty.}
 	\end{equation*}
+
+	The set of persistence diagrams together with any of the distances above is a
+	%\hyperref[metric space]{metric space}.


Even if this is commented out, I recommend using the correct label, i.e. "metric_space", inside the square brackets.

This seems still unaddressed, or am I missing something?

Still not showing up here :/

ulupo · 2020-04-05T19:31:55Z

doc/theory/glossary.tex

-	||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p}
-	\end{equation*}
-	whenever the right hand side exists and is finite.
+	%\hyperref[persistence diagram] {persistence diagram}


Even if this is commented out, I recommend using the correct label, i.e. "persistence_diagram", inside the square brackets.

ulupo · 2020-04-05T19:34:24Z

doc/theory/glossary.tex

+	We describe the graph of each $\lambda_k$ intuitively. For each $i \in I$, draw an isosceles triangle with base the interval $(b_i, d_i)$ on the horizontal $t$-axis, and sides with slope 1 and $-1$. This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If $P_k$ is the union of the polygonal regions with values at least $k$, then the graph of $\lambda_k$ is the upper contour of $P_k$, with $\lambda_k(a) = 0$ if the vertical line $t=a$ does not intersect $P_k$.
+
+	The persistence landscape construction defines a
+	%\hyperref[vectorization, kernel and amplitude]{vectorization}


Even if this is commented out, I recommend using the correct label, i.e. "vectorization_kernel_and_amplitude", inside the square brackets.

ulupo · 2020-04-05T19:35:24Z

doc/theory/glossary.tex

+	The persistence landscape construction defines a
+	%\hyperref[vectorization, kernel and amplitude]{vectorization}
+	vectorization of the set of persistence diagrams with target the vector space of real-valued function on $\mathbb N \times \mathbb R$. For any $p = 1,\dots,\infty$ we can restrict attention to persistence diagrams $D$ whose associated persistence landscape $\lambda$ is
+	%\hyperref[l^p norm]{$p$-integrable}


This label format won't work well in the future (when this will hopefully get un-commented) due to the presence of a white space. Maybe we can use "l^p_norm" or even just "lp_norm"?

ulupo · 2020-04-05T19:36:27Z

doc/theory/glossary.tex

+	||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p}	
+	\end{equation*}	
+	is finite. In this case we refer to \eqref{equation:persistence_landscape_norm} as the
+	%\hyperref[vectorization, kernel and amplitude]{amplitude}


Ditto as above.

ulupo · 2020-04-05T19:39:10Z

doc/theory/glossary.tex

-	% \hyperref[persistence_diagram] {persistence diagram}
-	persistence diagram. A \textit{weighted silhouette} associated to $D$ is a continuous function $\phi : \mathbb R \to \mathbb R$ of the form
+	Let $D = \{(b_i, d_i)\}_{i \in I}$ be a
+	%\hyperref[persistence diagram] {persistence diagram}


Ditto as above. Also there is here a whitespace between brackets and parentheses.

Signed-off-by: ammedmar <ammedmar@gmail.com>

ulupo · 2020-04-06T00:00:49Z

doc/theory/glossary.tex

@@ -307,76 +304,69 @@
 	\begin{equation*}
 	\sup_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_{\infty.}
 	\end{equation*}
+
+	The set of persistence diagrams together with any of the distances above is a
+	%\hyperref[metric space]{metric space}.


This seems still unaddressed, or am I missing something?

Signed-off-by: ammedmar <ammedmar@gmail.com>

ulupo · 2020-04-06T07:21:25Z

doc/theory/glossary.tex

@@ -307,76 +304,69 @@
 	\begin{equation*}
 	\sup_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_{\infty.}
 	\end{equation*}
+
+	The set of persistence diagrams together with any of the distances above is a
+	%\hyperref[metric space]{metric space}.


Still not showing up here :/

ulupo · 2020-04-06T07:22:12Z

doc/theory/glossary.tex

@@ -235,12 +232,12 @@
 	\subsection*{Vietoris-Rips complex and Vietoris-Rips persistence} \label{vietoris-rips_complex_and_vietoris-rips_persistence}

 	Let $(X, d)$ be a 
-	% \hyperref[finite_metric_spaces_and_point_clouds]{finite metric space}
+	% \hyperref[finite_metric_spaces_and_point_clouds]{finite metric_space}


Typo: "finite metric_space" should be "finite metric space" (displayed, not label name)

Great! Only this one still left...

ulupo · 2020-04-06T07:23:38Z

doc/theory/glossary.tex

-	||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p}
-	\end{equation*}
-	whenever the right hand side exists and is finite.
+	%\hyperref[persistence diagram] {persistence_diagram}


I think the underscore should be in the square brackets, not in the braces? Additionally, that whitespace is still showing up for me.

ulupo · 2020-04-06T07:24:28Z

doc/theory/glossary.tex

+	||u|| = \sqrt{\langle u, u \rangle}	
+	\end{equation*}	
+	and a norm space is naturally a
+	%\hyperref[metric_space]{metric_space}


There should be no underscore in the expression in braces I thought.

Signed-off-by: ammedmar <ammedmar@gmail.com>

ulupo · 2020-04-06T10:20:51Z

doc/theory/glossary.tex

@@ -235,12 +232,12 @@
 	\subsection*{Vietoris-Rips complex and Vietoris-Rips persistence} \label{vietoris-rips_complex_and_vietoris-rips_persistence}

 	Let $(X, d)$ be a 
-	% \hyperref[finite_metric_spaces_and_point_clouds]{finite metric space}
+	% \hyperref[finite_metric_spaces_and_point_clouds]{finite metric_space}


Great! Only this one still left...

Signed-off-by: ammedmar <ammedmar@gmail.com>

ulupo

Looks great!

* Add twine check to CI, attempt fix in README (#379) * Tests for plotting functions and methods, other increases in test coverage (#384) * Vectorization2 (#378) * Fix azure on windows (#388) * Allow more general input to ripser when metric='precomputed', improve/refactor check_point_clouds and add tests (#386) * Add citing entry to README.rst and doc/faq.rst (#389) * Fix markdown subsection formatting (#390) * Prepare release 0.2.1 (#391) Co-authored-by: Anibal M. Medina-Mardones <ammedmar@gmail.com> Co-authored-by: REDS institute <reds-heig@users.noreply.github.com> Co-authored-by: Guillaume Tauzin <guillaumetauzin.ut@gmail.com>

@wreise

* Add temporary patch to azure-pipelines.yml (#369) Get around Azure DevOps issues by following a suggestion by Sean Law: https://twitter.com/seanmylaw/status/1239332444865773568 * Remove metatransformers (#368) Signed-off-by: Guillaume Tauzin <guillaumetauzin.ut@gmail.com> * Extend plotting functionalities to more transformers and other modules (#362) * start plotting for images * Add the image plotting file * Add an imagePlotMixin subclassing the ImagePlot for theimage submodule, add persistenceImage plots * Add plot_persistence_image to the package * Add plotting functionalities to filtrations in the image module and forbetticurves * Preprocessing covered by plots * Add basic plotting functions for different vectorisation methods * Incorporate plotting for diagram representations and images/preprocessing * Change the docstring for ImagePlotterMixin and add fail on 3d images * Correct the samplingsin features, and titles * Small wording fix * Change dosctrings for plot in diagrams/features * Fix typo in variable name * Change plotting of heatkernel to the same as persistenceimage, use samplings * Cast images to floats, so as they can be always plotted * Add a Homology and PointCloud Mixin * Use the new PlotMixins * Add todo in Forgetdimensions, in relation to plotting * Add plotting to point_cloudsrescaling, correctimports * Wording changes * Correct n_values to n_bins and add 's' to n_homologydimensionS * Correct spelling mistakes * Change the name of ...PlotterMixin to ...Plotter * Make SlidingWindowEmbedding a PointCloudPlotter * Fix typoin import * Fix linting * Correct the way the distance matrices are displayed, and add the SlidingWindow.plot method * Change behaviour of transform_plot to return the 0-th sample * Docstring fixes * Create diagrams/representations.py, and migrate amplitude to features.py This change reflects the new documentation structure and the difference now drawn there between feature creation for diagrams (scalars) and vector representations/vectorizations. These changes are mostly invisible to the user. * Remove unused identity function * Rename HomologyPlotter as DiagramPlotter * Add homology_dimensions kwarg to DiagramPlotter, revise docstring * Small docstring fixes * Try adding triggers in the yaml * Remove excessive blank line * Revert "Change behaviour of transform_plot to return the 0-th sample" This reverts commit 2fa0209. * Fix docstring for PlotterMixin * Further docstring refinements in `PlotterMixin` * Remove `DiagramPlotter`, differentiate between implementations of `plot` Also make plot methods static in classes in `gtda.homology` and in ForgetDimensino * Correctly import PlotterMixin * Correct the plotting in the lorentz notebook * Remove PointCloudPlotter and refine plot methods for point clouds * Thorough refactoring of plotting API - Eliminate some top-level functions for diagram representations, leave only betti_curves and betti_surfaces - Remove ImagePlotter mixin and plotting/_plot.py, create top-level plot_heatmap function - Rename plotting/diagrams.py to plotting/diagrams_representations.py - Rename plotting/image.py to plotting/images.py - Rename plotting/homology.py to plotting/persistence_digrams.py - Fix problem with Binarizer coming from now-removed normalize parameter - Refactor notebooks to use plotting API - MIscellaneous small changes elsewhere The philosophy behind the file changes in gtda/plotting is that file names now reflect data structures and do not try to follow the names of subpackages like diagrams, image, etc. * Grammar * Fix default parameter for homology_dimension_ix in HeatKernel * Change plot_diagrams to allow ForgetDimension to work * Allow _subdiagrams to better handle infs of different kind * Add inline comment for future selves * Write check_diagrams docstring and make it convert to float, revert previous changes to _subdiagrams * Pass copy=False to astype in check_diagrams Co-authored-by: Guillaume Tauzin <guillaume.tauzin@epfl.ch> Co-authored-by: Guillaume Tauzin <guillaumetauzin.ut@gmail.com> Co-authored-by: Umberto Lupo <46537483+ulupo@users.noreply.github.com> Co-authored-by: Umberto <u.lupo@l2f.ch> * Glossary entry: distance, inner products & kernels (#370) * Update P landscapes * Add distances, inner products and kernels glossary entry * Update after W's comments Signed-off-by: ammedmar <ammedmar@gmail.com> Co-authored-by: ammedmar <anibal@medina-mardones.com> * Add reference for images submodules (#372) * Add reference for images submodules Signed-off-by: Guillaume Tauzin <guillaumetauzin.ut@gmail.com> * Move hera to submodule (#371) * delete hera repository * Add hera as submodule * Remove unnecessary branch for submodule * Add parameter max_bids_per_round when computing wasserstein * Fix an inconsistency with wasserstein implementation In Wasserstein, by default epsilon_common_ratio is equal 5, and it can be equal 0, which our bindings did not allow * Add test for python bindings of wasserstein Add method to retrieve what wasserstein consider to be infinity as an input * Add test for python bindings of bottleneck * Add fixes done by author after I contact him Signed-off-by: julian <julian.burellaperez@heig-vd.ch> * Miscellaneous refinements to GitHub pages (#363) * Remove code fetching logo as it is now stored in doc/images * Fix logo in examples directory * Partial rewriting of contributing page * Remove instructions for building documentation * Simplify path in Makefile * Remove documentation job in azure-pipelines.yml * Add @wreise to authors in doc/conf.py * Remove paper citation from FAQ * Rename Glossary to Theory Glossary * Linting in images subpackage * Add PlotterMixin to base.rst * Update FAQ * Improve README with pointers to GH pages * Implement @wreise's review suggestions * Fix links to svgs * Remove remaining mentions of plotting.py * Rename basic to tutorials and advanced to examples * Fixes in jupyter notebooks in examples folder * Add plotting in docs * Fixes in glossary * Correct the VBox display * Add TODO instead of empty reference in Cech docstring * Fix references in docstrings * Reintroduce scrollZoom in mapper_quickstart * Improve landing page and move Lewis' blog post to notebooks/tutorials.rst Co-authored-by: Wojciech Reise <reisewojtus@gmail.com> * Pcs different sizes (#318) * Add a check_point_clouds fct in utils/validation, and apply it throughout classes in ``homology/simplicial.py`` * Adapt and add tests * Make the output of ImageToPointCloud variable in size, and adjust the tests (modify the expected values) * Add mapper and utils to global __init__ * Revise use of check_array throughout * Linting and code clarity improvements throughout * Fix some incorrect links to GH pages * Revert to trivial slicing in transform_plot method of PlotterMixin, to cover case of list input Co-authored-by: Umberto <u.lupo@l2f.ch> * Release of v0.2.0 (#375) * Bump version to 0.2.0, add release notes * Improve LaTeX rendering in mapper_quickstart.ipynb * Add @rth as a contributor and code author for v0.2.0 (#376) * Add @rth to code authors and list of contributors for v0.2.0 * [WIP] Add twine check to CI, attempt fix in README (#379) * Add twine check to azure-pipelines.yml * Fix formatting and duplicate issues in README.rst * Tests for plotting functions and methods, other increases in test coverage (#384) * Remove images subpackage from code omitted from coverage * Add tests for Filtering class * Add tests in diagrams subpackage, linting - Rename test_features -> test_features_representations - Create parametrized dummy tests for fit_transform_plot method of transformers in representations.py - Improve structure and range of tests in test_preprocessing in a similar way - Create dummy tests of shape for BettiCurve and PersistenceLandscape - Linting and code style changes elsewhere * Add fit_transform_plot test for GraphGeodesicDistance * Add fit_transform_plot tests in gtda.homology * Fix typo in voids_on_the_plane notebook * Add fit_transform_plot tests for images subpackage * Add fit_transform_plot tests for point_clouds subpackage * Add test for SlidingWindow.plot * Linting * Exclude gtda.utils._doc and gtda.utils.intervals from coveragerc * Implement @wreise's suggestions - Rename some test functions - Implement a solution to the automatic tab opening problem, based on setting the default value of plotly.io.renderers in test files * Upload coverage report only from Azure macOS job * Exclude test folders from coverage reports * Vectorization2 (#378) * Update P landscapes * Add distances, inner products and kernels glossary entry Signed-off-by: ammedmar <anibal@medina-mardones.com> * Fix azure on windows (#388) * Update ccache * Update variable used to indicate boost headers location Official answer from boost devs: https://discourse.cmake.org/t/findboost-difference-between-boost-include-dir-and-boost-include-dirs/855 * Add install boost step in Azure CI, which downloads boost Signed-off-by: julian <julian.burellaperez@heig-vd.ch> * Allow more general input to ripser when metric='precomputed', improve/refactor check_point_clouds and add tests (#386) * Remove boolean mask on infinite values in VietorisRipsPersistence * Refactoring of check_point_clouds - Rename distance_matrix kwarg to distance_matrices - Add checks for square input when distance_matrices is set to True - More comprehensive/friendly error messages and warnings * Increase generality of ripser function when metric == 'precomputed' Sklearn's pairwise_distances is no longer called when metric='precomputed'. This is because pairwise_distances checks for positivity and finite input in that case, but this is unnecessarily restrictive for ripser. pairwise_distances also checks for square input so this is no longer checked in the case metric='precomputed'. But the ripser function is not meant for use in isolation. It is meant to be called by a VietorisRipsPersistence instance, and input validation can be offloaded to that class. In particular, 9bab872 introduces checks for square input. * Add tests for check_point_clouds * Refresh docstrings for VietorisRipsPersistence - Mention "adjacency matrices of weighted undirected graphs" as well as distance matrices - Mention compatibility with filtration in docstrings for fit and transform * Add citing entry to README.rst and doc/faq.rst (#389) * Add citation section in README * Add citation info to FAQ page in docs Signed-off-by: Guillaume Tauzin <guillaumetauzin.ut@gmail.com> Co-authored-by: Umberto Lupo <46537483+ulupo@users.noreply.github.com> * Fix markdown subsection formatting (#390) * Fix citing giotto-tda section * Fix Important link section * Prepare release 0.2.1 (#391) * Make the docs and push to a test branch in gtda-docs * Fix indentation Co-authored-by: Umberto Lupo <46537483+ulupo@users.noreply.github.com> Co-authored-by: Guillaume Tauzin <guillaumetauzin.ut@gmail.com> Co-authored-by: Guillaume Tauzin <guillaume.tauzin@epfl.ch> Co-authored-by: Umberto <u.lupo@l2f.ch> Co-authored-by: Anibal M. Medina-Mardones <ammedmar@gmail.com> Co-authored-by: ammedmar <anibal@medina-mardones.com> Co-authored-by: REDS institute <reds-heig@users.noreply.github.com>

ammedmar and others added 3 commits March 13, 2020 10:40

Update P landscapes

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Add distances, inner products and kernels glossary entry

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Remake vectorization changes

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Merge branch 'master' into master

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Fix typo

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Signed-off-by: ammedmar <ammedmar@gmail.com>

ammedmar requested a review from ulupo April 6, 2020 10:40

ulupo approved these changes Apr 6, 2020

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ulupo merged commit 805e912 into giotto-ai:master Apr 6, 2020


		Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by $1/\sqrt{2}$.
		We describe the graph of each $\lambda_k$ intuitively. For each $i \in I$, draw an isosceles triangle with base the interval $(b_i, d_i)$ on the horizontal $t$-axis, and sides with slope 1 and $-1$. This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If $P_k$ is the union of the polygonal regions with values at least $k$, then the graph of $\lambda_k$ is the upper contour of $P_k$, with $\lambda_k(a) = 0$ if the vertical line $t=a$ does not intersect $P_k$.

Vectorization2 #378

Vectorization2 #378

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ammedmar commented Mar 23, 2020

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ammedmar Mar 23, 2020 • edited

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ammedmar Mar 23, 2020 •

edited