From f77fca22d5bf339dedd7dbd7aa18db74ef0334b0 Mon Sep 17 00:00:00 2001 From: SEOKWOOPARK Date: Wed, 17 Jun 2026 19:02:50 +0000 Subject: [PATCH 1/8] Add markdown for Schur/matrix-free memory performance in /doc --- doc/memory_performance.md | 158 +++++++++++++++++++++++++++++++++++++ doc/memory_performance.png | Bin 0 -> 187620 bytes 2 files changed, 158 insertions(+) create mode 100644 doc/memory_performance.md create mode 100644 doc/memory_performance.png diff --git a/doc/memory_performance.md b/doc/memory_performance.md new file mode 100644 index 0000000..8ac3ff6 --- /dev/null +++ b/doc/memory_performance.md @@ -0,0 +1,158 @@ +# Improvement by Schur and Matrix-Free + +GPU-memory benchmark sweep for `bae`'s Bundle Adjustment optimizers. It runs a +set of [BAL](https://grail.cs.washington.edu/projects/bal/) problems through +several optimizer configurations, records peak and average GPU memory, and plots +memory usage against problem size. + +## Output Graph + +![Peak and average GPU memory vs. number of optimizable parameters, for the three optimizer configurations](memory_performance.png) + +*Peak (left) and average (right) GPU memory allocation across the three +optimizer configurations, plotted against problem size.* + +## Background: Schur complement & matrix-free + +### 1) Schur complement + +Each Bundle Adjustment step solves a large sparse linear system — the normal +equations `H Δx = g`, with `H = JᵀJ`. The unknowns split into two groups: +**camera** parameters (poses + intrinsics) and **3D point** parameters. Ordering +them that way gives `H` a 2×2 block structure (symbols match `Schur.step`): + +``` +[ U W ] [ Δc ] [ g_c ] +[ Wᵀ V ] [ Δp ] = [ g_p ] +``` + +- `U = J0ᵀ J0` — camera block +- `V = J1ᵀ J1` — point block, **block-diagonal**: each 3D point touches only its + own observations, so `V⁻¹` is cheap to apply +- `W = J0ᵀ J1` — camera–point coupling + +Because `V` is trivially invertible, the point unknowns can be eliminated, +leaving a much smaller system in the cameras only — the **Schur complement** +system: + +``` +S Δc = g_c − W V⁻¹ g_p , where S = U − W V⁻¹ Wᵀ +``` + +`bae` solves this reduced system for the camera update `Δc` with preconditioned +CG, then **back-substitutes** to recover the points: + +``` +Δp = V⁻¹ (g_p − Wᵀ Δc) +``` + + +### 2) Matrix-free + +The operator `S` can be handled two ways, which is what the `matrix_free_normal` +flag selects. The branch lives in [`bae/optim/optimizer.py`](../bae/optim/optimizer.py) +— inside `Schur.step` (and `LM.step` for the non-Schur path): + +- **`False` — normal path** (the *Schur, normal* condition) — the matrices are **materialized**: `W`, + `Wᵀ`, the product `W V⁻¹ Wᵀ`, and the assembled `S` are built explicitly as + sparse matrices and handed to the solver. +- **`True` — matrix-free path** (the *Schur, matrix-free* condition) — `S` (and even `W`) is **never + formed**. CG only needs the result of `S·x` for a vector `x`, so the operator + is applied on the fly as a chain of sparse matrix–vector products (`J0`, `J1`, + `V⁻¹`, …) inside `schur_matvec`. + +Iterative solvers like CG never inspect the matrix entries — they only multiply +the matrix by a vector — so the matrix-free form computes the **same** step while +**avoiding the memory of storing `W`, `Wᵀ`, and `S`**. That saved memory grows +with problem size, which is exactly what the plots show: the matrix-free +conditions (*Schur, matrix-free* and *LM, matrix-free*) sit well below +*Schur, normal*. + +The same flag also applies to the plain `LM` optimizer, where it swaps the +explicitly built `JᵀJ` matrix for the matrix-free `NormalMatVec` operator. + +## What it measures + +| Metric | Meaning | +| --- | --- | +| Number of parameters | The count of optimizable parameters: 3 per 3D point (xyz position) plus 7 per camera — i.e. 3 × (number of 3D points) + 7 × (number of cameras) | +| Peak GPU memory | The peak allocation in MiB — the max of the background sampler, `torch.cuda.max_memory_allocated`, and the Warp mempool high-water mark | +| Average GPU memory | The mean allocation in MiB over the run, sampled every 5 ms | + +GPU memory is the sum of **PyTorch** allocations +and the **Warp** memory pool captured by a +background `MemorySampler` thread with interval 5ms. + +## Conditions + +Optimizer | `matrix_free_normal` | Color | +--- | --- | --- | +`Schur` | `True` | green | +`Schur` | `False` | orange | +`LM` | `True` | red | + +The optimizer setup (shared by all conditions) is: + +- Strategy: `TrustRegion(up=2.0, down=0.5**4)` +- Solver: `PCG(tol=1e-4, maxiter=250)` +- `reject=20` +- Model: `Residual` from [`examples/schur.py`](../examples/schur.py), optimizing + camera params + 3D points. Intrinsics are included + (`OPTIMIZE_INTRINSICS = True` → 10 camera params per camera). + +This whole setup about the `TrustRegion` / `PCG` / `reject=20` configuration and the +`Residual` model is taken from the reference example +[`examples/schur.py`](../examples/schur.py). + +## How it works + +The three benchmark conditions all optimize the same reprojection `Residual` +model from [`examples/schur.py`](../examples/schur.py), and trace a path from the +`LM` baseline to `Schur` with matrix-free normal equations — each step lowers GPU +memory (the mechanism behind each is detailed in *Background* above): + +1. **`LM` → `Schur`** — instead of solving the full normal equations, `Schur` + eliminates the block-diagonal point block `V` and only solves the much + smaller camera-only *reduced* system. Fewer unknowns reach the CG solver and + the reduced system is better conditioned. +2. **`+ matrix-free`** — the reduced operator `S` is applied through + matrix–vector products instead of being materialized, so `W`, `Wᵀ`, and `S` + are never stored. This is where the bulk of the memory saving comes from. + +Measured on the largest benchmark problem (`venice/problem-1778`, ≈ 3.0 M +optimizable parameters): + +| Condition | Peak | Average | +| --- | --- | --- | +| LM, matrix-free | ~10.3 GiB | ~5.8 GiB | +| Schur, normal | ~9.9 GiB | ~7.7 GiB | +| Schur, matrix-free | **~5.8 GiB** | **~2.3 GiB** | + +So **Schur with matrix-free** uses about **43% less peak** memory than the +**LM (matrix-free)** baseline (~4.5 GiB less) and **~60% less average**. Notably, +**Schur with the normal (materialized) path** has the *highest* average of the +three — building `W`/`Wᵀ`/`S` explicitly keeps that memory occupied for the +entire solve — so it is the **matrix-free** half that makes Schur pay off, not +the complement alone. **Schur with matrix-free** also has the flattest +memory-vs-size trend (~2.0 vs ~3.6 MiB per 1 K parameters at peak), so the gap +keeps widening as problems grow. + +## Configuration + +Knobs near the top of the script: + +| Constant | Value | Purpose | +| --- | --- | --- | +| `TARGET_PROBLEMS` | ladybug / trafalgar / dubrovnik / venice list | Which BAL problems to sweep | +| `CONDITIONS` | 3 configs | Optimizer / matrix-free combinations to compare | +| `DEVICE` | `"cuda"` | Compute device | +| `NUM_ITERATIONS` | `20` | LM steps per run | +| `OPTIMIZE_INTRINSICS` | `True` | Include camera intrinsics (10 vs 7 camera params) | +| `MEMORY_SAMPLE_INTERVAL_S` | `0.005` | Background memory sampling interval | + +## Requirements + +- CUDA-capable GPU (the script asserts a Warp CUDA device is available) +- PyTorch, NumPy, Matplotlib, [Warp](https://github.com/NVIDIA/warp) (`warp-lang`) +- `bae` installed (see the repo [README](../README.md)); BAL problems are + downloaded/cached on first use via `datapipes.bal_loader.get_problem`. diff --git a/doc/memory_performance.png b/doc/memory_performance.png new file mode 100644 index 0000000000000000000000000000000000000000..6a2f6e24b548f1076803d7b1b14a9cccc70cb985 GIT binary patch literal 187620 zcmdSBcRbf``#%1pK}3`ynRzFak?bGD3>%WEG)QWJD-L zl90%r-{aDKe?H&){{8*?gI6r#CEZ5 z5@PaO`Hs1}pK?PAuQtI5AK_Hp$_6b(h~R^MfO zCT8HvX&?QG1-jqO?@L;i><(Pnc(qtf+bHc-kiN!;cw^u9j_;RBKYGNL)VlhCO{1lB z_`ysaAw;1m58Jc<=j%3Z6p{Xa z{VhdoPMkOq_T~uviWS0r292Lfd^bqmUWtG5Pj)nUceMSqC^{NnCUfiq-|HiiD>>0OHBmys9R1pk+`TqSA@(PjF0|Tn+>Pb4MPCad0&)*w* z`}TSfN5AAjVm_D9Dm*5J^NO?G$N#-)PF(s+Qe{o)h$JDnWCa1gNJ(SIW}xa z(7P@9`}c2^-kR9q_p%+G-`<1=dMYhGG_Q%B-s3A>3(OrGUR72`xoamT_q>hP%{X*z 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TtFA%}xd!E!)=~PeXRiJqwq&^< literal 0 HcmV?d00001 From df53e70a40b7ff4783530160b00a121114445f3f Mon Sep 17 00:00:00 2001 From: SEOKWOOPARK Date: Wed, 17 Jun 2026 19:21:08 +0000 Subject: [PATCH 2/8] The rejection parameter 20 -> 30 --- doc/memory_performance.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/doc/memory_performance.md b/doc/memory_performance.md index 8ac3ff6..98ccd47 100644 --- a/doc/memory_performance.md +++ b/doc/memory_performance.md @@ -95,12 +95,12 @@ The optimizer setup (shared by all conditions) is: - Strategy: `TrustRegion(up=2.0, down=0.5**4)` - Solver: `PCG(tol=1e-4, maxiter=250)` -- `reject=20` +- `reject=30` - Model: `Residual` from [`examples/schur.py`](../examples/schur.py), optimizing camera params + 3D points. Intrinsics are included (`OPTIMIZE_INTRINSICS = True` → 10 camera params per camera). -This whole setup about the `TrustRegion` / `PCG` / `reject=20` configuration and the +This whole setup about the `TrustRegion` / `PCG` / `reject=30` configuration and the `Residual` model is taken from the reference example [`examples/schur.py`](../examples/schur.py). From eea3ea60deaaaddaecc6a0885ac5853af6fba730 Mon Sep 17 00:00:00 2001 From: SEOKWOOPARK Date: Thu, 18 Jun 2026 21:58:40 +0000 Subject: [PATCH 3/8] Replace the number of camera parameter from 7 to 10 with the corresponding markdown and the graph image --- doc/memory_performance.md | 11 +++++++---- doc/memory_performance.png | Bin 187620 -> 187811 bytes 2 files changed, 7 insertions(+), 4 deletions(-) diff --git a/doc/memory_performance.md b/doc/memory_performance.md index 98ccd47..0b670a4 100644 --- a/doc/memory_performance.md +++ b/doc/memory_performance.md @@ -75,9 +75,11 @@ explicitly built `JᵀJ` matrix for the matrix-free `NormalMatVec` operator. | Metric | Meaning | | --- | --- | -| Number of parameters | The count of optimizable parameters: 3 per 3D point (xyz position) plus 7 per camera — i.e. 3 × (number of 3D points) + 7 × (number of cameras) | +| Number of parameters | The x-axis count: 3 per 3D point (xyz position) plus 10 per camera (7 SE(3) pose + 3 intrinsics f, k1, k2) — i.e. 3 × (number of 3D points) + 10 × (number of cameras). With `OPTIMIZE_INTRINSICS = True` all 10 camera parameters are optimized, so `num_params` counts `NUM_CAMERA_PARAMS = 10` per camera. | | Peak GPU memory | The peak allocation in MiB — the max of the background sampler, `torch.cuda.max_memory_allocated`, and the Warp mempool high-water mark | | Average GPU memory | The mean allocation in MiB over the run, sampled every 5 ms | +| Initial / final loss | Sum-of-squared-residuals loss (`model.loss`) before and after the 20 LM steps — a convergence sanity check (not plotted) | +| Per-pixel loss | Final mean squared reprojection error per observation (`least_square_error`) = final loss ÷ number of observations | GPU memory is the sum of **PyTorch** allocations and the **Warp** memory pool captured by a @@ -97,7 +99,7 @@ The optimizer setup (shared by all conditions) is: - Solver: `PCG(tol=1e-4, maxiter=250)` - `reject=30` - Model: `Residual` from [`examples/schur.py`](../examples/schur.py), optimizing - camera params + 3D points. Intrinsics are included + camera params + 3D points (**3 per point**: xyz). Intrinsics are included (`OPTIMIZE_INTRINSICS = True` → 10 camera params per camera). This whole setup about the `TrustRegion` / `PCG` / `reject=30` configuration and the @@ -129,7 +131,7 @@ optimizable parameters): | Schur, matrix-free | **~5.8 GiB** | **~2.3 GiB** | So **Schur with matrix-free** uses about **43% less peak** memory than the -**LM (matrix-free)** baseline (~4.5 GiB less) and **~60% less average**. Notably, +**LM (matrix-free)** baseline (~4.5 GiB less) and **~61% less average**. Notably, **Schur with the normal (materialized) path** has the *highest* average of the three — building `W`/`Wᵀ`/`S` explicitly keeps that memory occupied for the entire solve — so it is the **matrix-free** half that makes Schur pay off, not @@ -147,7 +149,8 @@ Knobs near the top of the script: | `CONDITIONS` | 3 configs | Optimizer / matrix-free combinations to compare | | `DEVICE` | `"cuda"` | Compute device | | `NUM_ITERATIONS` | `20` | LM steps per run | -| `OPTIMIZE_INTRINSICS` | `True` | Include camera intrinsics (10 vs 7 camera params) | +| `OPTIMIZE_INTRINSICS` | `True` | Optimize the 3 camera intrinsics → `NUM_CAMERA_PARAMS = 10` (7 pose + 3 intrinsics) | +| `POINT_PARAMS` | `3` | Parameters per 3D point (xyz position) | | `MEMORY_SAMPLE_INTERVAL_S` | `0.005` | Background memory sampling interval | ## Requirements diff --git a/doc/memory_performance.png b/doc/memory_performance.png index 6a2f6e24b548f1076803d7b1b14a9cccc70cb985..eb5dc126b89d6d64b7d710487aa21ce36427cd06 100644 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TtFA%}xd!E!)=~PeXRiJqwq&^< From c76792749136cdb68d6d49c569b7abcf94bf116d Mon Sep 17 00:00:00 2001 From: SEOKWOOPARK Date: Thu, 18 Jun 2026 22:09:52 +0000 Subject: [PATCH 4/8] Remove a necessary expression in Configuration section --- doc/memory_performance.md | 2 -- 1 file changed, 2 deletions(-) diff --git a/doc/memory_performance.md b/doc/memory_performance.md index 0b670a4..9810b4b 100644 --- a/doc/memory_performance.md +++ b/doc/memory_performance.md @@ -141,8 +141,6 @@ keeps widening as problems grow. ## Configuration -Knobs near the top of the script: - | Constant | Value | Purpose | | --- | --- | --- | | `TARGET_PROBLEMS` | ladybug / trafalgar / dubrovnik / venice list | Which BAL problems to sweep | From 6c30f0f022021fb3305247bac10e458ba19681a9 Mon Sep 17 00:00:00 2001 From: SEOKWOOPARK Date: Fri, 19 Jun 2026 04:16:20 +0000 Subject: [PATCH 5/8] Change the title of the markdown focusing on neutral comparison --- doc/memory_performance.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/memory_performance.md b/doc/memory_performance.md index 9810b4b..cac1f90 100644 --- a/doc/memory_performance.md +++ b/doc/memory_performance.md @@ -1,4 +1,4 @@ -# Improvement by Schur and Matrix-Free +# Comparing GPU Memory: Schur, Matrix-Free, and LM GPU-memory benchmark sweep for `bae`'s Bundle Adjustment optimizers. It runs a set of [BAL](https://grail.cs.washington.edu/projects/bal/) problems through From 0bbe4214a0adf9afd8d0698d09c284068e112e1f Mon Sep 17 00:00:00 2001 From: SEOKWOOPARK Date: Fri, 19 Jun 2026 04:35:27 +0000 Subject: [PATCH 6/8] Change the title of the markdown focusing on neutral comparison --- doc/memory_performance.md | 3 --- 1 file changed, 3 deletions(-) diff --git a/doc/memory_performance.md b/doc/memory_performance.md index cac1f90..e17d66d 100644 --- a/doc/memory_performance.md +++ b/doc/memory_performance.md @@ -39,9 +39,6 @@ system: S Δc = g_c − W V⁻¹ g_p , where S = U − W V⁻¹ Wᵀ ``` -`bae` solves this reduced system for the camera update `Δc` with preconditioned -CG, then **back-substitutes** to recover the points: - ``` Δp = V⁻¹ (g_p − Wᵀ Δc) ``` From 7dac7e7b71c4355abd519d0256a1014f9f7c6548 Mon Sep 17 00:00:00 2001 From: SEOKWOOPARK Date: Fri, 19 Jun 2026 05:06:40 +0000 Subject: [PATCH 7/8] Add the context of what Bundle Adjustment optimizers mean --- doc/memory_performance.md | 26 +++++++++++++++++++++----- 1 file changed, 21 insertions(+), 5 deletions(-) diff --git a/doc/memory_performance.md b/doc/memory_performance.md index e17d66d..74886ff 100644 --- a/doc/memory_performance.md +++ b/doc/memory_performance.md @@ -1,9 +1,25 @@ # Comparing GPU Memory: Schur, Matrix-Free, and LM -GPU-memory benchmark sweep for `bae`'s Bundle Adjustment optimizers. It runs a -set of [BAL](https://grail.cs.washington.edu/projects/bal/) problems through -several optimizer configurations, records peak and average GPU memory, and plots -memory usage against problem size. +**Bundle Adjustment (BA)** is the nonlinear least-squares problem at the core of +3D reconstruction, Structure-from-Motion and SLAM. It jointly refines camera poses +and intrinsics together with 3D point positions to minimize reprojection error. +As scenes scale to millions of parameters, GPU +memory becomes the bottleneck, and how that linear system is +solved largely decides how much memory is used. + +This page benchmarks GPU memory across the two implementation choices `bae` +offers for that solve: + +- **`LM` vs `Schur`** — solve the full normal equations (`LM`, + Levenberg–Marquardt) or use the **Schur complement** to reduce them to the + cameras only (`Schur`). +- **matrix-free vs materialized** — apply the operator through matrix–vector + products meaning matrix-free or build it explicitly in memory. + +The sweep runs a set of [BAL](https://grail.cs.washington.edu/projects/bal/) +problems of increasing size through three configurations —> `Schur` matrix-free, +`Schur` materialized, and `LM` matrix-free. This records peak and average GPU +memory and plots it against problem size. ## Output Graph @@ -140,7 +156,7 @@ keeps widening as problems grow. | Constant | Value | Purpose | | --- | --- | --- | -| `TARGET_PROBLEMS` | ladybug / trafalgar / dubrovnik / venice list | Which BAL problems to sweep | +| `TARGET_PROBLEMS` | ladybug / trafalgar / dubrovnik / venice list | target datasets | | `CONDITIONS` | 3 configs | Optimizer / matrix-free combinations to compare | | `DEVICE` | `"cuda"` | Compute device | | `NUM_ITERATIONS` | `20` | LM steps per run | From a088bb46dc2d8eb2a4252200074bb911696d6ff6 Mon Sep 17 00:00:00 2001 From: SEOKWOOPARK Date: Sat, 20 Jun 2026 20:52:25 +0000 Subject: [PATCH 8/8] Change the title of the markdown and remove BA's definition --- doc/memory_performance.md | 9 +-------- 1 file changed, 1 insertion(+), 8 deletions(-) diff --git a/doc/memory_performance.md b/doc/memory_performance.md index 74886ff..d8c783c 100644 --- a/doc/memory_performance.md +++ b/doc/memory_performance.md @@ -1,11 +1,4 @@ -# Comparing GPU Memory: Schur, Matrix-Free, and LM - -**Bundle Adjustment (BA)** is the nonlinear least-squares problem at the core of -3D reconstruction, Structure-from-Motion and SLAM. It jointly refines camera poses -and intrinsics together with 3D point positions to minimize reprojection error. -As scenes scale to millions of parameters, GPU -memory becomes the bottleneck, and how that linear system is -solved largely decides how much memory is used. +# Comparing GPU Memory: Vanilla LM, Schur-complement Optimizer, and Matrix-free options This page benchmarks GPU memory across the two implementation choices `bae` offers for that solve: