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Rmd e50c51b Matthew Stephens 2026-08-13 Add BM experiments with multiple planted vectors, unbalanced groups,

Introduction

I want to experiment with the Burer-Monteiro approach to solving an SDP.

My overall interest is finding a v to maximize v’Pv subject to v is binary (-1,1). The classic SDP approach is to replace this with maximize tr(PV) subject to diag(V)=1 and V is psd. Then to take a random vector r and split the points according to Vr.

The BM approach avoids solving the full SDP by assuming V=YY’ for some nxr matrix Y where r is small-ish. Then optimize Y by gradient-based methods. According to Gemini, the theory suggests choosing r=sqrt(n) but smaller r may work well.

In my case P=UU’ for some orthogonal U, so we want to maximize tr(UU’YY’) subject to diag(YY’)=1 using a gradient-based method.

I’ll use Claude to help me write a function to do this.

We compare this approach with two other approaches: i) the simple heuristic of using \(Y\) to be the row-normalized \(U\). This is effectively trying to solve the split using random projections onto the column space of \(U\).ii) fastICA.

The objective is \(\text{tr}(UU'YY') = \|U'Y\|_F^2\), and the constraint \(\text{diag}(YY')=1\) means each row of \(Y\) lies on the unit sphere. This is Riemannian optimization on the product of \(n\) unit spheres. Note that the BM objective is not a true upper bound on the binary problem: binary solutions \(v\) are feasible for BM (set \(Y = v\), since \(|v_i|=1\)), so the global BM optimum \(\geq\) binary optimum, but gradient ascent only finds local optima and can fall below the best binary solution found by rounding.

The BM approach

This was written by Claude.

The Euclidean gradient is \(\nabla_Y f = 2UU'Y\). The Riemannian gradient for row \(i\) subtracts the normal component: \(\tilde{g}_i = g_i - (g_i \cdot y_i) y_i\). After a gradient step, we retract back to the sphere by normalizing each row.

After finding \(Y\), we round to a binary vector via random hyperplane rounding: draw \(g \sim N(0,I_r)\) and return \(v = \text{sign}(Yg)\).

#' Burer-Monteiro approach to maximizing v'UU'v subject to v in {-1,1}^n
#'
#' @param U n x k orthonormal matrix (P = UU')
#' @param r rank of the BM factorization (default: ceil(sqrt(n)))
#' @param n_iter number of gradient ascent iterations
#' @param step_size Riemannian gradient ascent step size
#' @param seed random seed for initialization
#' @return list with Y (n x r factor), obj_history, and SDP objective value
burer_monteiro_fit <- function(U, r = NULL, n_iter = 2000, step_size = 0.05, seed = 1) {
  n <- nrow(U)
  k <- ncol(U)
  if (is.null(r)) r <- max(2L, ceiling(sqrt(n)))

  set.seed(seed)
  # Initialize Y with rows on unit sphere
  Y <- matrix(rnorm(n * r), n, r)
  Y <- Y / sqrt(rowSums(Y^2))

  obj_history <- numeric(n_iter)

  for (iter in seq_len(n_iter)) {
    # Euclidean gradient: 2 * UU' Y  (compute as U (U'Y) for efficiency)
    UtY  <- crossprod(U, Y)      # k x r
    grad <- 2 * (U %*% UtY)     # n x r  =  2 UU'Y

    # Riemannian gradient: project out normal (row-wise)
    dots  <- rowSums(grad * Y)   # n-vector: <grad_i, y_i>
    rgrad <- grad - dots * Y     # n x r

    # Gradient ascent + retraction (normalize rows)
    Y <- Y + step_size * rgrad
    Y <- Y / sqrt(rowSums(Y^2))

    # Objective: tr(UU' YY') = ||U'Y||_F^2
    UtY <- crossprod(U, Y)
    obj_history[iter] <- sum(UtY^2)
  }

  list(Y = Y, obj_history = obj_history, sdp_obj = tail(obj_history, 1))
}

#' Round BM solution Y to binary vector via random hyperplane rounding
#'
#' @param Y n x r matrix from burer_monteiro_fit
#' @param U n x k matrix (to evaluate objective)
#' @param n_rounds number of random hyperplanes to try
#' @param seed random seed
#' @return list with v (best binary vector) and obj (v'UU'v)
bm_round <- function(Y, U, n_rounds = 200, seed = 42) {
  set.seed(seed)
  r <- ncol(Y)
  best_v   <- NULL
  best_obj <- -Inf

  best_x <- NULL
  best_w <- NULL
  for (i in seq_len(n_rounds)) {
    w   <- rnorm(r)
    x   <- drop(Y %*% w)
    v   <- sign(x)
    obj <- sum(crossprod(U, v)^2)   # ||U'v||^2 = v'Pv
    if (obj > best_obj) {
      best_obj <- obj
      best_v   <- v
      best_x   <- x
      best_w   <- w
    }
  }
  list(v = best_v, x = best_x, w=best_w, obj = best_obj)
}

#' Run fastICA n_runs times and return the binarized component with the best v'Pv objective
fastica_best <- function(U, n_comp, n_runs = 10) {
  best_v   <- NULL
  best_obj <- -Inf
  for (run in seq_len(n_runs)) {
    fit <- fastICA::fastICA(U, n.comp = n_comp)
    for (j in seq_len(n_comp)) {
      v_j   <- sign(fit$S[, j])
      obj_j <- sum(crossprod(U, v_j)^2)
      if (obj_j > best_obj) { best_obj <- obj_j; best_v <- v_j }
    }
  }
  list(v = best_v, obj = best_obj)
}

Simple test

Generate a random \(P = UU'\), run BM, and compare the rounded solution to a brute-force search over random binary vectors.

The first issue is to generate a random U with a planted binary (-1,1) vector. In initial explorations it seems that exactly how you do this can affect the results quite a lot, and this might be worth understanding better. Here I generate the first column of U as a planted vector and the remaining columns as a uniform at random among vectors orthogonal to the planted vector.

# Simulate U with n_plant planted binary vectors (need not be orthogonal to each other).
# P = UU' projects onto the span of the planted vectors plus k-n_plant noise directions,
# so every planted vector achieves v'Pv = n (the maximum).
sim_U = function(n, k, seed, n_plant = 1, p = 0.5) {
  set.seed(seed)

  # 1. Generate n_plant random ±1 vectors and find an orthonormal basis for their span
  V_raw   <- matrix(sample(c(-1, 1), n * n_plant, replace = TRUE, prob = c(1-p, p)), n, n_plant)
  V       <- V_raw / rep(sqrt(colSums(V_raw^2)), each = n)
  planted <- qr.Q(qr(V))                    # n x rank(V); handles collinear vectors

  m_eff <- ncol(planted)

  # 2. Random noise for remaining k - m_eff dimensions, projected off the planted span
  noise      <- matrix(rnorm(n * (k - m_eff)), n, k - m_eff)
  noise_orth <- noise - planted %*% crossprod(planted, noise)
  U_noise    <- qr.Q(qr(noise_orth))

  list(U = cbind(planted, U_noise), v_true = V_raw)
}

Small \(k\) (\(k=5\))

First we try small k so everything should work:

n <- 100
k <- 5
sim    <- sim_U(n, k, seed = 1)
U      <- sim$U
v_truth <- sim$v_true[, 1]


# Baseline: round random projections of row-normalized U
U_rownorm <- U / sqrt(rowSums(U^2))
baseline  <- bm_round(U_rownorm, U, n_rounds = 5000)
brute_best_v <- baseline$v
brute_obj    <- baseline$obj

# Burer-Monteiro
fit <- burer_monteiro_fit(U, r = 8, n_iter = 3000, step_size = 0.05)
rnd <- bm_round(fit$Y, U, n_rounds = 500)


# Convergence plot
plot(fit$obj_history, type = "l", xlab = "Iteration", ylab = "SDP objective",
     main = "BM convergence")

#fastICA
fica     <- fastica_best(U, n_comp = k)
v_fica   <- fica$v
obj_fica <- fica$obj

obj_truth <- sum(crossprod(U, v_truth)^2)

knitr::kable(data.frame(
  Method    = c("Planted truth", "BM rounded", "Baseline (row-norm U)", "fastICA binarized", "BM continuous (Y)"),
  Objective = round(c(obj_truth, rnd$obj, brute_obj, obj_fica, fit$sdp_obj), 2),
  Cor_truth = round(c(1, cor(rnd$v, v_truth), cor(brute_best_v, v_truth), abs(cor(v_fica, v_truth)), NA), 3)
), col.names = c("Method", "Objective", "Cor. with truth"))
Method Objective Cor. with truth
Planted truth 100 1
BM rounded 100 1
Baseline (row-norm U) 100 1
fastICA binarized 100 1
BM continuous (Y) 100 NA

Moderate \(k\) (\(k=20\))

Now try larger \(k = 20\). Here the BM method solves the SDP adequately (it achieves the same objective as the planted truth) but the rounded solution is almost orthogonal to the truth, and achieves only 85% of the optimal objective. In contrast, fastICA (run 10 times here) finds the optimal.

n <- 100
k <- 20
sim     <- sim_U(n, k, seed = 1)
U       <- sim$U
v_truth <- sim$v_true[, 1]


# Baseline: round random projections of row-normalized U
U_rownorm <- U / sqrt(rowSums(U^2))
baseline  <- bm_round(U_rownorm, U, n_rounds = 5000)
brute_best_v <- baseline$v
brute_obj    <- baseline$obj

# Burer-Monteiro
fit <- burer_monteiro_fit(U, r = 8, n_iter = 3000, step_size = 0.05)
rnd <- bm_round(fit$Y, U, n_rounds = 500)


# Convergence plot
plot(fit$obj_history, type = "l", xlab = "Iteration", ylab = "SDP objective",
     main = "BM convergence")

#fastICA
fica     <- fastica_best(U, n_comp = k)
v_fica   <- fica$v
obj_fica <- fica$obj

obj_truth <- sum(crossprod(U, v_truth)^2)

knitr::kable(data.frame(
  Method    = c("Planted truth", "BM rounded", "Baseline (row-norm U)", "fastICA binarized", "BM continuous (Y)"),
  Objective = round(c(obj_truth, rnd$obj, brute_obj, obj_fica, fit$sdp_obj), 2),
  Cor_truth = round(c(1, cor(rnd$v, v_truth), cor(brute_best_v, v_truth), abs(cor(v_fica, v_truth)), NA), 3)
), col.names = c("Method", "Objective", "Cor. with truth"))
Method Objective Cor. with truth
Planted truth 100.00 1.000
BM rounded 84.33 -0.820
Baseline (row-norm U) 82.72 0.079
fastICA binarized 100.00 1.000
BM continuous (Y) 99.98 NA

Now try larger \(n=1000, k=20\); all the methods work. (Here fastICA only needs running once)

n <- 1000
k= 20
sim     <- sim_U(n, k, seed = 1)
U       <- sim$U
v_truth <- sim$v_true[, 1]


# Baseline: round random projections of row-normalized U
U_rownorm <- U / sqrt(rowSums(U^2))
baseline  <- bm_round(U_rownorm, U, n_rounds = 5000)
brute_best_v <- baseline$v
brute_obj    <- baseline$obj

# Burer-Monteiro
fit <- burer_monteiro_fit(U, r = 8, n_iter = 3000, step_size = 0.05)
rnd <- bm_round(fit$Y, U, n_rounds = 500)


# Convergence plot
plot(fit$obj_history, type = "l", xlab = "Iteration", ylab = "SDP objective",
     main = "BM convergence")

#fastICA
fica     <- fastica_best(U, n_comp = k, n_runs=1)
v_fica   <- fica$v
obj_fica <- fica$obj

obj_truth <- sum(crossprod(U, v_truth)^2)

knitr::kable(data.frame(
  Method    = c("Planted truth", "BM rounded", "Baseline (row-norm U)", "fastICA binarized", "BM continuous (Y)"),
  Objective = round(c(obj_truth, rnd$obj, brute_obj, obj_fica, fit$sdp_obj), 2),
  Cor_truth = round(c(1, cor(rnd$v, v_truth), cor(brute_best_v, v_truth), abs(cor(v_fica, v_truth)), NA), 3)
), col.names = c("Method", "Objective", "Cor. with truth"))
Method Objective Cor. with truth
Planted truth 1000.00 1.00
BM rounded 1000.00 -1.00
Baseline (row-norm U) 726.91 -0.73
fastICA binarized 1000.00 1.00
BM continuous (Y) 999.97 NA

Larger \(k\) (\(k=60\))

Now try larger \(k = 60\), \(n=1000\). (I pushed it to 60 to make the BM method not work) Here I had to increase r in the BM method to get it to find an objective that matches the known optimal. I found that the fastICA method sometimes found the truth; with this seed I had to increase the repeats to 20 to get it to find the truth.

n <- 1000
k= 60
sim     <- sim_U(n, k, seed = 1)
U       <- sim$U
v_truth <- sim$v_true[, 1]


# Baseline: round random projections of row-normalized U
U_rownorm <- U / sqrt(rowSums(U^2))
baseline  <- bm_round(U_rownorm, U, n_rounds = 5000)
brute_best_v <- baseline$v
brute_obj    <- baseline$obj

# Burer-Monteiro
fit <- burer_monteiro_fit(U, r = 30, n_iter = 3000, step_size = 0.05)
rnd <- bm_round(fit$Y, U, n_rounds = 500)


# Convergence plot
plot(fit$obj_history, type = "l", xlab = "Iteration", ylab = "SDP objective",
     main = "BM convergence")

#fastICA
fica     <- fastica_best(U, n_comp = k, n_runs=20)
v_fica   <- fica$v
obj_fica <- fica$obj

obj_truth <- sum(crossprod(U, v_truth)^2)

knitr::kable(data.frame(
  Method    = c("Planted truth", "BM rounded", "Baseline (row-norm U)", "fastICA binarized", "BM continuous (Y)"),
  Objective = round(c(obj_truth, rnd$obj, brute_obj, obj_fica, fit$sdp_obj), 2),
  Cor_truth = round(c(1, cor(rnd$v, v_truth), cor(brute_best_v, v_truth), abs(cor(v_fica, v_truth)), NA), 3)
), col.names = c("Method", "Objective", "Cor. with truth"))
Method Objective Cor. with truth
Planted truth 1000.00 1.000
BM rounded 712.86 0.662
Baseline (row-norm U) 696.56 -0.103
fastICA binarized 1000.00 1.000
BM continuous (Y) 999.67 NA

Now I’ll run fastICA initializing from the BM solution (which here is already 61% correlated with the truth). Interestingly, it does not move much.

fastica_r1update = function(X,w){
  w= w/sqrt(sum(w^2))
  P = t(X) %*% w
  G = tanh(P)
  G2 = 1-tanh(P)^2
  w = X %*% G - sum(G2) * w 
  return(w)
}

w = t(U) %*% rnd$x
for(i in 1:10000){
  w = fastica_r1update(t(U),w) 
}

cor(U %*% w, v_truth)
          [,1]
[1,] 0.6917269

Multiple vectors

I’m interested in the case where \(U\) contains multiple different binary vectors. What I want to do is find them as local optima of some objective. I’m interested in whether the BM approach will find both vectors.

Simulate U with first two columns planted:

n <- 1000; k <- 60
sim    <- sim_U(n, k, seed = 2, n_plant = 2)
U      <- sim$U
v_true <- sim$v_true   # n x 2 matrix of planted ±1 vectors
best_cor_planted <- function(v, v_true) max(abs(cor(v, v_true)))

# Baseline
U_rownorm    <- U / sqrt(rowSums(U^2))
baseline     <- bm_round(U_rownorm, U, n_rounds = 5000)

# Burer-Monteiro
fit <- burer_monteiro_fit(U, r = 30, n_iter = 3000, step_size = 0.05)
rnd <- bm_round(fit$Y, U, n_rounds = 500)

# fastICA
fica     <- fastica_best(U, n_comp = k)
v_fica   <- fica$v
obj_fica <- fica$obj

knitr::kable(data.frame(
  Method     = c("BM rounded", "Baseline (row-norm U)", "fastICA binarized", "BM continuous (Y)"),
  Objective  = round(c(rnd$obj, baseline$obj, obj_fica, fit$sdp_obj), 2),
  Best_cor   = round(c(best_cor_planted(rnd$v,       v_true),
                       best_cor_planted(baseline$v,   v_true),
                       best_cor_planted(v_fica,        v_true),
                       NA), 3)
), col.names = c("Method", "Objective", "Best cor. with planted"))
Method Objective Best cor. with planted
BM rounded 708.85 0.520
Baseline (row-norm U) 694.19 0.188
fastICA binarized 1000.00 1.000
BM continuous (Y) 999.76 NA

Unbalanced groups

Since unbalanced groups can be more difficult to find I try that case here. I simulate U with one planted column, with 0.8 -1 and 0.2 +1 (i.e. p=0.2), with \(n=1000\), \(k=20\). Here BM works but ICA does not, because (as we know) ICA does not work well for unbalanced groups. It seems the SPD approach still works for unbalanced groups (which makes sense since it was designed with max cut in mind, and so it would be a fatal flaw if it didn’t work for unbalanced cuts)

n <- 1000; k <- 20
sim    <- sim_U(n, k, seed = 1, n_plant = 1, p = 0.2)
U      <- sim$U
v_true <- sim$v_true

best_cor_planted <- function(v, v_true) max(abs(cor(v, v_true)))

# Baseline
U_rownorm <- U / sqrt(rowSums(U^2))
baseline  <- bm_round(U_rownorm, U, n_rounds = 5000)

# Burer-Monteiro
fit <- burer_monteiro_fit(U, r = 8, n_iter = 3000, step_size = 0.05)
rnd <- bm_round(fit$Y, U, n_rounds = 500)

# fastICA
fica     <- fastica_best(U, n_comp = k)
v_fica   <- fica$v
obj_fica <- fica$obj

knitr::kable(data.frame(
  Method    = c("BM rounded", "Baseline (row-norm U)", "fastICA binarized", "BM continuous (Y)"),
  Objective = round(c(rnd$obj, baseline$obj, obj_fica, fit$sdp_obj), 2),
  Best_cor  = round(c(best_cor_planted(rnd$v,      v_true),
                      best_cor_planted(baseline$v,  v_true),
                      best_cor_planted(v_fica,       v_true),
                      NA), 3)
), col.names = c("Method", "Objective", "Best cor. with planted"))
Method Objective Best cor. with planted
BM rounded 1000.00 1.000
Baseline (row-norm U) 729.05 0.656
fastICA binarized 709.46 0.024
BM continuous (Y) 999.97 NA 

sessionInfo()
R version 4.4.2 (2024-10-31)
Platform: aarch64-apple-darwin20
Running under: macOS 26.5.2

Matrix products: default
BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0

locale:
[1] C

time zone: America/Chicago
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] vctrs_0.7.2     cli_3.6.5       knitr_1.51      rlang_1.1.7    
 [5] xfun_0.56       stringi_1.8.7   otel_0.2.0      promises_1.5.0 
 [9] jsonlite_2.0.0  workflowr_1.7.2 glue_1.8.0      rprojroot_2.1.1
[13] git2r_0.36.2    htmltools_0.5.9 httpuv_1.6.16   sass_0.4.10    
[17] rmarkdown_2.30  fastICA_1.2-7   evaluate_1.0.5  jquerylib_0.1.4
[21] tibble_3.3.1    fastmap_1.2.0   yaml_2.3.12     lifecycle_1.0.5
[25] whisker_0.4.1   stringr_1.6.0   compiler_4.4.2  fs_1.6.6       
[29] Rcpp_1.1.1      pkgconfig_2.0.3 later_1.4.6     digest_0.6.39  
[33] R6_2.6.1        pillar_1.11.1   magrittr_2.0.4  bslib_0.10.0   
[37] tools_4.4.2     cachem_1.1.0