introduction to topological data analysismvj/gc-bigdata... · 4/18/2019 · •category theory:...
TRANSCRIPT
![Page 1: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/1.jpg)
Mikael Vejdemo-Johansson
Introduction toTopological Data Analysis
![Page 2: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/2.jpg)
• Data has shape
• Homology: linear algebra measures shape
• Persistence: squinting with mathematics
• Cohomology
• Applications
Outline
![Page 3: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/3.jpg)
• Data has shape
• Homology: linear algebra measures shape
• Persistence: squinting with mathematics
• Cohomology
• Applications
Outline
![Page 4: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/4.jpg)
Data has shape
![Page 5: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/5.jpg)
Data has shape
![Page 6: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/6.jpg)
Data has shape
![Page 7: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/7.jpg)
Data has shape
![Page 8: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/8.jpg)
Data has shape
![Page 9: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/9.jpg)
• Data has shape
• Homology: linear algebra measures shape
• Persistence: squinting with mathematics
• Cohomology
• Applications
Outline
![Page 10: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/10.jpg)
![Page 11: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/11.jpg)
![Page 12: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/12.jpg)
C0
![Page 13: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/13.jpg)
C1
![Page 14: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/14.jpg)
C2
![Page 15: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/15.jpg)
Linear maps C2 C1
- +
C1 C0
-
∂:
∂:
![Page 16: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/16.jpg)
Boundary of paths telescope
∂ =
-
-
-
+
+
+
![Page 17: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/17.jpg)
Boundary of paths telescope
∂ =
-
-
-
+
+
+
![Page 18: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/18.jpg)
Boundary of paths telescope
∂ =
-
-
+
+
![Page 19: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/19.jpg)
Boundary of paths telescope
∂ =
-
-
+
+
![Page 20: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/20.jpg)
Boundary of paths telescope
∂ =
-
Note: if the path z is a cycle, endpoints coincide, so ∂z=0
![Page 21: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/21.jpg)
• A chain is a linear combination of simplices
• A cycle is an element of ker ∂ Something that looks like a closed path
• A boundary is an element of img ∂ Something that should look like a closed path
• Homology is the quotient vector space ker ∂ / img ∂ Essential (non-obvious) cycles
Definitions
![Page 22: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/22.jpg)
• Data has shape
• Homology: linear algebra measures shape
• Persistence: squinting with mathematics
• Cohomology
• Applications
Outline
![Page 23: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/23.jpg)
![Page 24: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/24.jpg)
![Page 25: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/25.jpg)
![Page 26: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/26.jpg)
![Page 27: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/27.jpg)
![Page 28: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/28.jpg)
![Page 29: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/29.jpg)
![Page 30: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/30.jpg)
![Page 31: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/31.jpg)
![Page 32: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/32.jpg)
3x intersection
![Page 33: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/33.jpg)
3x intersection
![Page 34: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/34.jpg)
3x intersection
![Page 35: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/35.jpg)
![Page 36: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/36.jpg)
![Page 37: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/37.jpg)
![Page 38: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/38.jpg)
![Page 39: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/39.jpg)
![Page 40: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/40.jpg)
What parameter to pick?
![Page 41: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/41.jpg)
• Look at all parameters at once
• Scale-independent
• Compressed data summary
• As the parameter grows, complexes include Inclusions create linear maps on homology
• Track homology classes as the parameter growsHomology is born, lives, and dies.
Persistent homology
![Page 42: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/42.jpg)
• Category theory: Homology is a functor.Sequence of spaces X1 → X2 → X3 → X4.Maps to sequence of vector spaces with induced linear maps:HkX1 → HkX2 → HkX3 → HkX4.
• Module theory (over 𝕜[t] or over an An quiver or…): Sequence of vector spaces and linear maps decomposes into direct sum of interval modules.These are defined by birth index and death index.
Persistent homology
![Page 43: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/43.jpg)
Persistent barcodesPersistent diagrams
![Page 44: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/44.jpg)
Persistent barcodesPersistent diagrams
![Page 45: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/45.jpg)
• Data has shape
• Homology: linear algebra measures shape
• Persistence: squinting with mathematics
• Cohomology
• Applications
Outline
![Page 46: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/46.jpg)
• Instead of chains Cn, use cochains Cn=Hom(Cn,𝕜)
• Instead of boundary ∂ use coboundary δ=∂T
• Cocycles: follow a generalized path invariance:f(x,y) + f(y,z) = f(x,z) for any paths x→y, x→z, y→z
• Coboundaries: path invariance follows from a generalized potential function construction:f = g(y)-g(x)
Cohomology: dualize everything
![Page 47: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/47.jpg)
• Fun fact:H1(X; ℤ) is bijective with homotopy equivalence classes of functions X → S1
• Construction for [f]∈ H1(X; ℤ):
• Send vertices to 0
• Use f(e) as a wrapping number: wrap e around the circle f(e) times
• All higher simplices work out bc path invariance
1-Cohomology isCircle-valued functions
![Page 48: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/48.jpg)
• Construction adapts to data: Pretend [f] is instead from H1(X; ℝ).arg ming | f - δg |2 mod 1.0is a smooth function C0 → S1.
• Can compute intrinsic phase variables from geometry of time series
1-Cohomology isCircle-valued functions
![Page 49: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/49.jpg)
• Data has shape
• Homology: linear algebra measures shape
• Persistence: squinting with mathematics
• Cohomology
• Applications
Outline
![Page 50: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/50.jpg)
![Page 51: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/51.jpg)
![Page 52: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/52.jpg)
![Page 53: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/53.jpg)
![Page 54: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/54.jpg)
![Page 55: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/55.jpg)
![Page 56: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/56.jpg)
![Page 57: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/57.jpg)
![Page 58: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/58.jpg)
![Page 59: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/59.jpg)
![Page 60: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/60.jpg)
![Page 61: Introduction to Topological Data Analysismvj/GC-BigData... · 4/18/2019 · •Category theory: Homology is a functor. Sequence of spaces X1 → X2 → X3 → X4. Maps to sequence](https://reader034.vdocument.in/reader034/viewer/2022050305/5f6d98bdaf761462313b4d4c/html5/thumbnails/61.jpg)