[BUG] Accuracy of lanczos solver.

[trivialfis]

This is a tracking issue for the accuracy of Lanczos solver. During the development of the spectral embedding interface for cuml, I compared the eigen solver between raft and scipy eigsh with a normalized Laplacian matrix as input. The result from raft solver is off for some cases. At the end, I have pasted a 16x16 example COO matrix extracted out from spectral embedding.

Related: rapidsai/cuml#3864 rapidsai/cuml#263

  std::vector<int32_t> rows{
    0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  1,  1,
    1,  1,  1,  1,  1,  1,  2,  2,  2,  2,  2,  2,  2,  2,  3,  3,  3,  3,  3,
    3,  3,  3,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  5,  5,  5,  5,  5,  5,
    5,  5,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  7,  7,  7,  7,  7,  7,
    7,  7,  7,  7,  7,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,
    8,  9,  9,  9,  9,  9,  9,  9,  9,  10, 10, 10, 10, 10, 10, 10, 10, 11, 11,
    11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12,
    13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14,
    14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15};
  std::vector<int32_t> cols{
    13, 11, 10, 9,  5, 3,  2,  1,  14, 15, 6,  12, 4,  7,  8,  0,  3,  9,  7,
    14, 8,  0,  11, 2, 1,  12, 0,  1,  8,  3,  9,  11, 2,  1,  0,  12, 8,  2,
    11, 9,  3,  7,  5, 14, 8,  13, 0,  15, 12, 6,  4,  4,  7,  0,  6,  10, 15,
    14, 5,  10, 7,  5, 8,  13, 0,  12, 14, 4,  15, 6,  10, 5,  1,  12, 4,  8,
    6,  14, 15, 0,  7, 13, 11, 10, 9,  3,  2,  1,  15, 7,  14, 6,  4,  12, 0,
    8,  1,  0,  12, 8, 2,  11, 3,  9,  8,  0,  7,  6,  15, 14, 5,  10, 1,  0,
    12, 8,  9,  3,  2, 11, 11, 9,  7,  3,  2,  14, 15, 6,  8,  13, 0,  4,  12,
    14, 15, 8,  0,  6, 12, 4,  13, 13, 12, 10, 1,  8,  7,  4,  0,  5,  6,  15,
    14, 13, 12, 10, 5, 8,  7,  0,  4,  14, 6,  15};
  std::vector<float> vals{-0.067419983446598053, -0.064282432198524475, -0.07537783682346344, -0.064282432198524475, -0.067419983446598053, -0.067419983446598053, -0.064282432198524475, -0.064282432198524475, -0.10050377994775772, -0.10341754555702209, -0.10341754555702209, -0.10341754555702209, -0.10660035908222198, -0.11396057903766632, -0.093048423528671265,    1, -0.095346257090568542, -0.18181817233562469, -0.080582298338413239, -0.071066908538341522, -0.065795168280601501, -0.064282432198524475, -0.18181817233562469, -0.18181817233562469,    1, -0.073127247393131256, -0.064282432198524475, -0.18181817233562469, -0.065795168280601501, -0.19069251418113708, -0.18181817233562469, -0.18181817233562469,    1, -0.095346257090568542, -0.067419983446598053, -0.076696500182151794, -0.069006554782390594, -0.19069251418113708, -0.19069251418113708, -0.19069251418113708,    1, -0.066815309226512909, -0.079056940972805023, -0.1178511306643486, -0.10910894721746445, -0.15811388194561005, -0.10660035163164139, -0.12126781791448593, -0.12126781791448593, -0.12126781791448593,    1, -0.079056940972805023, -0.08451543003320694, -0.067419983446598053, -0.076696500182151794, -0.22360679507255554, -0.15339300036430359, -0.14907120168209076,    1, -0.085749298334121704, -0.064820379018783569, -0.076696500182151794, -0.10585123300552368, -0.15339300036430359, -0.10341754555702209, -0.11764707416296005, -0.1143324002623558, -0.12126782536506653, -0.11764707416296005,    1, -0.094491124153137207, -0.08451543003320694, -0.080582298338413239, -0.064820379018783569, -0.066815316677093506, -0.11664237827062607, -0.064820379018783569, -0.12598817050457001, -0.12964075803756714, -0.11396057903766632,    1, -0.069006554782390594, -0.065795168280601501, -0.077151678502559662, -0.065795168280601501, -0.069006554782390594, -0.065795168280601501, -0.065795168280601501, -0.10585123300552368, -0.11664237827062607, -0.10286890715360641, -0.10585123300552368, -0.10910895466804504, -0.10585123300552368, -0.093048423528671265,    1, -0.18181817233562469, -0.064282432198524475, -0.073127247393131256, -0.065795168280601501, -0.18181817233562469, -0.18181817233562469, -0.19069251418113708,    1, -0.077151678502559662, -0.07537783682346344, -0.094491124153137207, -0.085749298334121704, -0.085749298334121704, -0.083333335816860199, -0.22360679507255554,    1, -0.18181817233562469, -0.064282432198524475, -0.073127247393131256, -0.065795168280601501, -0.18181817233562469, -0.19069251418113708, -0.18181817233562469,    1, -0.073127247393131256, -0.073127247393131256, -0.064820379018783569, -0.076696500182151794, -0.073127247393131256, -0.057166200131177902, -0.058823537081480026, -0.11764707416296005, -0.10585123300552368, -0.15339300036430359, -0.10341754555702209, -0.12126782536506653,    1, -0.07453560084104538, -0.076696500182151794, -0.069006554782390594, -0.067419983446598053, -0.15339300036430359, -0.15339300036430359, -0.15811388194561005,    1, -0.07453560084104538, -0.057166200131177902, -0.083333335816860199, -0.071066908538341522, -0.10286890715360641, -0.12598817050457001, -0.1178511381149292, -0.10050377994775772, -0.14907120168209076, -0.1143324002623558, -0.1143324002623558,    1, -0.076696500182151794, -0.058823537081480026, -0.085749298334121704, -0.15339300036430359, -0.10585123300552368, -0.12964075803756714, -0.10341754555702209, -0.12126782536506653, -0.1143324002623558, -0.11764707416296005,    1};

[trivialfis]

cc @dantegd @cjnolet

[github-actions]

This issue has been labeled inactive-90d due to no recent activity in the past 90 days. Please close this issue if no further response or action is needed. Otherwise, please respond with a comment indicating any updates or changes to the original issue and/or confirm this issue still needs to be addressed.

[github-actions]

This issue has been labeled inactive-30d due to no recent activity in the past 30 days. Please close this issue if no further response or action is needed. Otherwise, please respond with a comment indicating any updates or changes to the original issue and/or confirm this issue still needs to be addressed. This issue will be labeled inactive-90d if there is no activity in the next 60 days.

[trivialfis]

I'm working on it.

[github-actions]

This issue has been labeled inactive-30d due to no recent activity in the past 30 days. Please close this issue if no further response or action is needed. Otherwise, please respond with a comment indicating any updates or changes to the original issue and/or confirm this issue still needs to be addressed. This issue will be labeled inactive-90d if there is no activity in the next 60 days.

[cjnolet]

Linking #379

[github-actions]

This issue has been labeled inactive-30d due to no recent activity in the past 30 days. Please close this issue if no further response or action is needed. Otherwise, please respond with a comment indicating any updates or changes to the original issue and/or confirm this issue still needs to be addressed. This issue will be labeled inactive-90d if there is no activity in the next 60 days.

[github-actions]

This issue has been labeled inactive-90d due to no recent activity in the past 90 days. Please close this issue if no further response or action is needed. Otherwise, please respond with a comment indicating any updates or changes to the original issue and/or confirm this issue still needs to be addressed.

[cjnolet]

@trivialfis is this something we might be able to prioritize for 22.08? There is also a plan to start integrating lbfgs into cupy so it might make for a good theme to do some improvements to our solvers in 22.08/2.10 timeframe.

[github-actions]

This issue has been labeled inactive-30d due to no recent activity in the past 30 days. Please close this issue if no further response or action is needed. Otherwise, please respond with a comment indicating any updates or changes to the original issue and/or confirm this issue still needs to be addressed. This issue will be labeled inactive-90d if there is no activity in the next 60 days.