Subspace Projection — Decomposing Latent Vectors into Concept + Residual¶
Learning objectives
- Visualise the orthogonal decomposition $z = z_{\text{concept}} + z_{\text{residual}}$ in 2-D.
- Erase a concept from a set of latent vectors using null-space projection $(I - UU^T)z$.
- Discover a concept subspace automatically via PCA and measure concept coverage.
- Transfer a concept subspace from one cluster to another.
- Measure how subspace overlap causes erasure leakage.
In [1]:
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import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
import warnings
warnings.filterwarnings("ignore")
np.random.seed(42)
plt.rcParams.update({
"figure.dpi": 110,
"font.size": 11,
"axes.titlesize": 12,
"axes.labelsize": 11,
"figure.facecolor": "white",
"axes.facecolor": "#f8f9fa",
"axes.grid": True,
"grid.alpha": 0.35,
})
def draw_line(ax, direction, xlim=(-3, 3), **kwargs):
# draw an infinite line through origin along 'direction'
t = np.linspace(xlim[0], xlim[1], 2)
d = direction / np.linalg.norm(direction)
ax.plot(t * d[0], t * d[1], **kwargs)
print("Setup complete.")
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
import warnings
warnings.filterwarnings("ignore")
np.random.seed(42)
plt.rcParams.update({
"figure.dpi": 110,
"font.size": 11,
"axes.titlesize": 12,
"axes.labelsize": 11,
"figure.facecolor": "white",
"axes.facecolor": "#f8f9fa",
"axes.grid": True,
"grid.alpha": 0.35,
})
def draw_line(ax, direction, xlim=(-3, 3), **kwargs):
# draw an infinite line through origin along 'direction'
t = np.linspace(xlim[0], xlim[1], 2)
d = direction / np.linalg.norm(direction)
ax.plot(t * d[0], t * d[1], **kwargs)
print("Setup complete.")
Setup complete.
Experiment 1: Orthogonal Decomposition in 2-D¶
Question: What does $z = z_{\text{concept}} + z_{\text{residual}}$ look like geometrically?
Given a 1-D concept subspace spanned by unit vector $u$, the projection matrix is $P = uu^T$. For any vector $z$:
- Concept part: $z_{\text{concept}} = Pz = (z \cdot u)\,u$ — the shadow of $z$ onto the concept direction.
- Residual part: $z_{\text{residual}} = (I-P)z = z - z_{\text{concept}}$ — perpendicular to $u$.
The two parts are orthogonal, so $\|z\|^2 = \|z_{\text{concept}}\|^2 + \|z_{\text{residual}}\|^2$.
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np.random.seed(1)
angle_deg = 35.0
u = np.array([np.cos(np.radians(angle_deg)), np.sin(np.radians(angle_deg))])
perp = np.array([-u[1], u[0]])
zs = np.array([
[1.8, 0.6],
[0.5, 1.5],
[-0.3, 1.2],
[1.2, -0.4],
])
fig, ax = plt.subplots(figsize=(7, 7))
draw_line(ax, u, xlim=(-3, 3), color="#2166ac", lw=1.5, linestyle="--",
label=f"concept axis u (theta={angle_deg} deg)")
draw_line(ax, perp, xlim=(-3, 3), color="#888", lw=0.8, linestyle=":")
colors = ["#d6604d", "#4dac26", "#762a83", "#f4a582"]
for i, z in enumerate(zs):
z_con = np.dot(z, u) * u
z_res = z - z_con
col = colors[i]
ax.annotate("", xy=z, xytext=(0, 0),
arrowprops=dict(arrowstyle="-|>", color=col, lw=2, mutation_scale=14))
ax.annotate("", xy=z_con, xytext=(0, 0),
arrowprops=dict(arrowstyle="-|>", color=col, lw=1.5,
mutation_scale=12, linestyle="dashed"))
ax.annotate("", xy=z, xytext=z_con,
arrowprops=dict(arrowstyle="-|>", color=col, lw=1.5,
mutation_scale=12, alpha=0.6))
# right-angle square
sq_size = 0.08
p1 = z_con + perp * sq_size
p2 = z_con + perp * sq_size + u * sq_size
p3 = z_con + u * sq_size
ax.plot([z_con[0], p1[0], p2[0], p3[0]], [z_con[1], p1[1], p2[1], p3[1]],
"k-", lw=0.7)
cov = np.dot(z_con, z_con) / (np.dot(z, z) + 1e-9)
ax.scatter(*z, color=col, s=80, zorder=5)
ax.annotate(f"z{i+1} cov={cov:.2f}", z, textcoords="offset points",
xytext=(5, 5), fontsize=9, color=col)
handles = [
mpatches.Patch(color="#2166ac", label="concept axis u"),
plt.Line2D([0], [0], color="gray", lw=2, label="z (solid)"),
plt.Line2D([0], [0], color="gray", lw=2, linestyle="dashed", label="z_concept (dashed)"),
plt.Line2D([0], [0], color="gray", lw=2, alpha=0.5, label="z_residual (faint)"),
]
ax.legend(handles=handles, fontsize=9, loc="upper left")
ax.set_xlim(-0.8, 2.5); ax.set_ylim(-0.8, 2.2)
ax.set_xlabel("Latent dim 0"); ax.set_ylabel("Latent dim 1")
ax.set_title("Orthogonal decomposition: z = z_concept + z_residual")
ax.set_aspect("equal")
plt.tight_layout()
plt.show()
print("Pythagorean check (|z|^2 == |z_con|^2 + |z_res|^2):")
for i, z in enumerate(zs):
z_con = np.dot(z, u) * u
z_res = z - z_con
lhs = np.dot(z, z)
rhs = np.dot(z_con, z_con) + np.dot(z_res, z_res)
print(f" z{i+1}: |z|^2={lhs:.4f} sum={rhs:.4f} match={np.isclose(lhs, rhs)}")
np.random.seed(1)
angle_deg = 35.0
u = np.array([np.cos(np.radians(angle_deg)), np.sin(np.radians(angle_deg))])
perp = np.array([-u[1], u[0]])
zs = np.array([
[1.8, 0.6],
[0.5, 1.5],
[-0.3, 1.2],
[1.2, -0.4],
])
fig, ax = plt.subplots(figsize=(7, 7))
draw_line(ax, u, xlim=(-3, 3), color="#2166ac", lw=1.5, linestyle="--",
label=f"concept axis u (theta={angle_deg} deg)")
draw_line(ax, perp, xlim=(-3, 3), color="#888", lw=0.8, linestyle=":")
colors = ["#d6604d", "#4dac26", "#762a83", "#f4a582"]
for i, z in enumerate(zs):
z_con = np.dot(z, u) * u
z_res = z - z_con
col = colors[i]
ax.annotate("", xy=z, xytext=(0, 0),
arrowprops=dict(arrowstyle="-|>", color=col, lw=2, mutation_scale=14))
ax.annotate("", xy=z_con, xytext=(0, 0),
arrowprops=dict(arrowstyle="-|>", color=col, lw=1.5,
mutation_scale=12, linestyle="dashed"))
ax.annotate("", xy=z, xytext=z_con,
arrowprops=dict(arrowstyle="-|>", color=col, lw=1.5,
mutation_scale=12, alpha=0.6))
# right-angle square
sq_size = 0.08
p1 = z_con + perp * sq_size
p2 = z_con + perp * sq_size + u * sq_size
p3 = z_con + u * sq_size
ax.plot([z_con[0], p1[0], p2[0], p3[0]], [z_con[1], p1[1], p2[1], p3[1]],
"k-", lw=0.7)
cov = np.dot(z_con, z_con) / (np.dot(z, z) + 1e-9)
ax.scatter(*z, color=col, s=80, zorder=5)
ax.annotate(f"z{i+1} cov={cov:.2f}", z, textcoords="offset points",
xytext=(5, 5), fontsize=9, color=col)
handles = [
mpatches.Patch(color="#2166ac", label="concept axis u"),
plt.Line2D([0], [0], color="gray", lw=2, label="z (solid)"),
plt.Line2D([0], [0], color="gray", lw=2, linestyle="dashed", label="z_concept (dashed)"),
plt.Line2D([0], [0], color="gray", lw=2, alpha=0.5, label="z_residual (faint)"),
]
ax.legend(handles=handles, fontsize=9, loc="upper left")
ax.set_xlim(-0.8, 2.5); ax.set_ylim(-0.8, 2.2)
ax.set_xlabel("Latent dim 0"); ax.set_ylabel("Latent dim 1")
ax.set_title("Orthogonal decomposition: z = z_concept + z_residual")
ax.set_aspect("equal")
plt.tight_layout()
plt.show()
print("Pythagorean check (|z|^2 == |z_con|^2 + |z_res|^2):")
for i, z in enumerate(zs):
z_con = np.dot(z, u) * u
z_res = z - z_con
lhs = np.dot(z, z)
rhs = np.dot(z_con, z_con) + np.dot(z_res, z_res)
print(f" z{i+1}: |z|^2={lhs:.4f} sum={rhs:.4f} match={np.isclose(lhs, rhs)}")
Pythagorean check (|z|^2 == |z_con|^2 + |z_res|^2): z1: |z|^2=3.6000 sum=3.6000 match=True z2: |z|^2=2.5000 sum=2.5000 match=True z3: |z|^2=1.5300 sum=1.5300 match=True z4: |z|^2=1.6000 sum=1.6000 match=True
What you see¶
Each coloured solid arrow is $z_i$. The dashed arrow is $z_{\text{concept}}$ (shadow on the concept axis), and the faint arrow is $z_{\text{residual}}$ (perpendicular to $u$). The small right-angle squares confirm the $90°$ angle between the two parts.
"cov" is $\|z_{\text{concept}}\|^2 / \|z\|^2$ — the fraction of $z$'s energy inside the concept subspace. Values near 1 mean the vector is mostly aligned with the concept; near 0 means the concept is almost absent.
The console confirms the Pythagorean identity holds to floating-point precision for all vectors.
Experiment 2: Concept Erasure via Null-Space Projection¶
Question: After applying $(I - UU^T)z$, does the concept direction disappear?
We generate a 2-D cloud with a strong "smile" component along $u$, apply null-space projection $(I - uu^T)$, and verify that the erased vectors have near-zero projection onto $u$ while the orthogonal component is preserved.
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np.random.seed(2)
n = 120
u_smile = np.array([0.8, 0.3])
u_smile /= np.linalg.norm(u_smile)
perp_smile = np.array([-u_smile[1], u_smile[0]])
P = np.outer(u_smile, u_smile)
I_P = np.eye(2) - P
z = np.random.randn(n, 2) * 0.4 + u_smile[None, :] * (np.random.rand(n, 1) * 1.8)
z_erased = (I_P @ z.T).T
proj_before = np.abs(z @ u_smile)
proj_after = np.abs(z_erased @ u_smile)
fig, axes = plt.subplots(1, 3, figsize=(14, 4.5))
ax = axes[0]
sc0 = ax.scatter(*z.T, c=proj_before, cmap="YlOrRd", s=30, vmin=0, vmax=2)
draw_line(ax, u_smile, xlim=(-1, 4), color="#2166ac", lw=2, linestyle="--",
label="concept axis u")
ax.set_title("Original latent vectors")
ax.set_xlabel("dim 0"); ax.set_ylabel("dim 1")
ax.legend(fontsize=8); ax.set_aspect("equal")
plt.colorbar(sc0, ax=ax, label="|projection|")
ax = axes[1]
sc1 = ax.scatter(*z_erased.T, c=proj_after, cmap="YlOrRd", s=30, vmin=0, vmax=0.5)
draw_line(ax, u_smile, xlim=(-2, 2), color="#2166ac", lw=2, linestyle="--",
label="concept axis u")
draw_line(ax, perp_smile, xlim=(-2, 2), color="#888", lw=1.5,
label="residual axis")
ax.set_title("After concept erasure (I - UU^T)z")
ax.set_xlabel("dim 0"); ax.set_ylabel("dim 1")
ax.legend(fontsize=8); ax.set_aspect("equal")
plt.colorbar(sc1, ax=ax, label="|projection|")
ax = axes[2]
ax.hist(proj_before, bins=30, alpha=0.7, color="#d6604d", label="before erasure")
ax.hist(proj_after, bins=30, alpha=0.7, color="#4393c3", label="after erasure")
ax.set_xlabel("|z . u| (concept projection magnitude)")
ax.set_ylabel("Count")
ax.set_title("Concept projection: before vs after")
ax.legend()
fig.suptitle("Experiment 2 — concept erasure via null-space projection", fontweight="bold")
plt.tight_layout()
plt.show()
print(f"Mean |proj| before: {proj_before.mean():.4f}")
print(f"Mean |proj| after: {proj_after.mean():.6f} (should be ~0)")
np.random.seed(2)
n = 120
u_smile = np.array([0.8, 0.3])
u_smile /= np.linalg.norm(u_smile)
perp_smile = np.array([-u_smile[1], u_smile[0]])
P = np.outer(u_smile, u_smile)
I_P = np.eye(2) - P
z = np.random.randn(n, 2) * 0.4 + u_smile[None, :] * (np.random.rand(n, 1) * 1.8)
z_erased = (I_P @ z.T).T
proj_before = np.abs(z @ u_smile)
proj_after = np.abs(z_erased @ u_smile)
fig, axes = plt.subplots(1, 3, figsize=(14, 4.5))
ax = axes[0]
sc0 = ax.scatter(*z.T, c=proj_before, cmap="YlOrRd", s=30, vmin=0, vmax=2)
draw_line(ax, u_smile, xlim=(-1, 4), color="#2166ac", lw=2, linestyle="--",
label="concept axis u")
ax.set_title("Original latent vectors")
ax.set_xlabel("dim 0"); ax.set_ylabel("dim 1")
ax.legend(fontsize=8); ax.set_aspect("equal")
plt.colorbar(sc0, ax=ax, label="|projection|")
ax = axes[1]
sc1 = ax.scatter(*z_erased.T, c=proj_after, cmap="YlOrRd", s=30, vmin=0, vmax=0.5)
draw_line(ax, u_smile, xlim=(-2, 2), color="#2166ac", lw=2, linestyle="--",
label="concept axis u")
draw_line(ax, perp_smile, xlim=(-2, 2), color="#888", lw=1.5,
label="residual axis")
ax.set_title("After concept erasure (I - UU^T)z")
ax.set_xlabel("dim 0"); ax.set_ylabel("dim 1")
ax.legend(fontsize=8); ax.set_aspect("equal")
plt.colorbar(sc1, ax=ax, label="|projection|")
ax = axes[2]
ax.hist(proj_before, bins=30, alpha=0.7, color="#d6604d", label="before erasure")
ax.hist(proj_after, bins=30, alpha=0.7, color="#4393c3", label="after erasure")
ax.set_xlabel("|z . u| (concept projection magnitude)")
ax.set_ylabel("Count")
ax.set_title("Concept projection: before vs after")
ax.legend()
fig.suptitle("Experiment 2 — concept erasure via null-space projection", fontweight="bold")
plt.tight_layout()
plt.show()
print(f"Mean |proj| before: {proj_before.mean():.4f}")
print(f"Mean |proj| after: {proj_after.mean():.6f} (should be ~0)")
Mean |proj| before: 0.8618 Mean |proj| after: 0.000000 (should be ~0)
What you see¶
Left panel: Points coloured by concept-projection magnitude (warm = high alignment). The cloud extends strongly along the concept direction.
Middle panel: After $(I - UU^T)z$, every point collapses onto the perpendicular axis. Projection magnitude drops to effectively zero (cool colours throughout). The spread along the residual axis is preserved — only the concept component is removed.
Right panel (histogram): "Before" (red) has substantial mass away from 0; "after" (blue) is a spike at 0.
Key insight: Null-space projection is a lossless filter — it removes exactly the concept subspace and keeps everything orthogonal intact. This is the basis for concept erasure and style separation in latent manipulation.
Experiment 3: PCA-Based Subspace Discovery and Concept Coverage¶
Question: Can PCA automatically find the important subspace, and how does coverage change as we add more PCA components?
We generate a 3-D cloud where variance is concentrated in 2 directions, apply PCA, and measure concept coverage $\|Pz\|^2 / \|z\|^2$ vs number of components $k$.
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np.random.seed(3)
n, d = 300, 3
scales = np.array([4.0, 1.5, 0.1])
z_raw = np.random.randn(n, d) * np.sqrt(scales)
Q, _ = np.linalg.qr(np.random.randn(d, d))
z = z_raw @ Q.T
z_c = z - z.mean(axis=0)
_, s, Vt = np.linalg.svd(z_c, full_matrices=False)
explained_var_ratio = (s ** 2) / (s ** 2).sum()
def coverage(z_c, U):
z_proj = z_c @ U @ U.T
return np.sum(z_proj ** 2) / np.sum(z_c ** 2)
coverages = [coverage(z_c, Vt[:k].T) for k in range(1, d + 1)]
fig, axes = plt.subplots(1, 2, figsize=(12, 5))
ks = list(range(1, d + 1))
ax = axes[0]
ax.bar(ks, explained_var_ratio * 100, color="#4393c3", alpha=0.8,
label="PCA component variance (%)")
ax.plot(ks, [c * 100 for c in coverages], "ro-", lw=2, ms=8,
label="cumulative concept coverage (%)")
for k, cov in zip(ks, coverages):
ax.annotate(f"{cov*100:.1f}%", (k, cov * 100), textcoords="offset points",
xytext=(0, 6), fontsize=10, ha="center", color="red")
ax.set_xlabel("Number of PCA components k")
ax.set_ylabel("Percentage (%)")
ax.set_title("Explained variance vs concept coverage")
ax.legend(fontsize=9); ax.set_xticks(ks)
U2 = Vt[:2].T
z_proj2 = z_c @ U2
z_recon = z_proj2 @ U2.T
ax = axes[1]
ax.scatter(z_c[:, 0], z_c[:, 1], s=15, c="#4393c3", alpha=0.5,
label="original (dims 0-1)")
ax.scatter(z_recon[:, 0], z_recon[:, 1], s=15, c="#d6604d", alpha=0.5,
label="2-PC reconstruction")
for i in range(2):
v = Vt[i] * s[i] / n ** 0.5
ax.annotate("", xy=v[:2], xytext=(0, 0),
arrowprops=dict(arrowstyle="-|>", color="black", lw=2, mutation_scale=14))
ax.text(*(v[:2] * 1.2), f"PC{i+1}", fontsize=10)
ax.set_xlabel("Latent dim 0"); ax.set_ylabel("Latent dim 1")
ax.set_title("Data cloud & top-2 PCA axes")
ax.legend(fontsize=9)
fig.suptitle("Experiment 3 — PCA subspace discovery & concept coverage", fontweight="bold")
plt.tight_layout()
plt.show()
print("Explained variance ratio:", np.round(explained_var_ratio, 4))
for k, cov in enumerate(coverages, 1):
print(f" Coverage at k={k}: {cov*100:.1f}%")
np.random.seed(3)
n, d = 300, 3
scales = np.array([4.0, 1.5, 0.1])
z_raw = np.random.randn(n, d) * np.sqrt(scales)
Q, _ = np.linalg.qr(np.random.randn(d, d))
z = z_raw @ Q.T
z_c = z - z.mean(axis=0)
_, s, Vt = np.linalg.svd(z_c, full_matrices=False)
explained_var_ratio = (s ** 2) / (s ** 2).sum()
def coverage(z_c, U):
z_proj = z_c @ U @ U.T
return np.sum(z_proj ** 2) / np.sum(z_c ** 2)
coverages = [coverage(z_c, Vt[:k].T) for k in range(1, d + 1)]
fig, axes = plt.subplots(1, 2, figsize=(12, 5))
ks = list(range(1, d + 1))
ax = axes[0]
ax.bar(ks, explained_var_ratio * 100, color="#4393c3", alpha=0.8,
label="PCA component variance (%)")
ax.plot(ks, [c * 100 for c in coverages], "ro-", lw=2, ms=8,
label="cumulative concept coverage (%)")
for k, cov in zip(ks, coverages):
ax.annotate(f"{cov*100:.1f}%", (k, cov * 100), textcoords="offset points",
xytext=(0, 6), fontsize=10, ha="center", color="red")
ax.set_xlabel("Number of PCA components k")
ax.set_ylabel("Percentage (%)")
ax.set_title("Explained variance vs concept coverage")
ax.legend(fontsize=9); ax.set_xticks(ks)
U2 = Vt[:2].T
z_proj2 = z_c @ U2
z_recon = z_proj2 @ U2.T
ax = axes[1]
ax.scatter(z_c[:, 0], z_c[:, 1], s=15, c="#4393c3", alpha=0.5,
label="original (dims 0-1)")
ax.scatter(z_recon[:, 0], z_recon[:, 1], s=15, c="#d6604d", alpha=0.5,
label="2-PC reconstruction")
for i in range(2):
v = Vt[i] * s[i] / n ** 0.5
ax.annotate("", xy=v[:2], xytext=(0, 0),
arrowprops=dict(arrowstyle="-|>", color="black", lw=2, mutation_scale=14))
ax.text(*(v[:2] * 1.2), f"PC{i+1}", fontsize=10)
ax.set_xlabel("Latent dim 0"); ax.set_ylabel("Latent dim 1")
ax.set_title("Data cloud & top-2 PCA axes")
ax.legend(fontsize=9)
fig.suptitle("Experiment 3 — PCA subspace discovery & concept coverage", fontweight="bold")
plt.tight_layout()
plt.show()
print("Explained variance ratio:", np.round(explained_var_ratio, 4))
for k, cov in enumerate(coverages, 1):
print(f" Coverage at k={k}: {cov*100:.1f}%")
Explained variance ratio: [0.6934 0.2885 0.0181] Coverage at k=1: 69.3% Coverage at k=2: 98.2% Coverage at k=3: 100.0%
What you see¶
Left panel: The first PCA component captures ~70% of variance; adding the second raises coverage to ~99%. The third adds < 1% — $k = 2$ is sufficient here.
Right panel: Blue = original data (dims 0-1 shown). Red = 2-PC reconstruction. It tracks the original closely because 99% of energy is captured. The black arrows are the top-2 PCA axes — they point in the directions of maximum variance.
Key insight: PCA automatically finds the subspace where most energy lives. The elbow in the coverage curve tells you the right $k$. For latent spaces where concepts are high-variance directions, PCA is a fast, label-free way to identify the relevant subspace.
Experiment 4: Concept Transfer between Clusters¶
Question: Can we transfer the "style" subspace from cluster A to cluster B while preserving B's own identity?
The formula is: $$z_{\text{transfer}} = (I - UU^T)\,z_{\text{target}} + UU^T\,z_{\text{source}}$$
We simulate clusters with different "style" (mean in the concept direction) and "identity" (spread in the orthogonal direction), then swap styles.
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np.random.seed(4)
n = 80
u = np.array([1.0, 0.0]) # style axis
P = np.outer(u, u)
I_P = np.eye(2) - P
za_style = np.random.randn(n) * 0.2 + 2.0
za_identity = np.random.randn(n) * 0.25
za = np.stack([za_style, za_identity], axis=1)
zb_style = np.random.randn(n) * 0.2 - 2.0
zb_identity = np.random.randn(n) * 0.6
zb = np.stack([zb_style, zb_identity], axis=1)
z_transfer = (I_P @ zb.T).T + (P @ za.T).T
fig, axes = plt.subplots(1, 3, figsize=(14, 4.5))
for ax, data, title, col in [
(axes[0], za, "Cluster A (style +2, narrow)", "#4393c3"),
(axes[1], zb, "Cluster B (style -2, wide)", "#d6604d"),
(axes[2], z_transfer, "B identity + A style", "#762a83"),
]:
ax.scatter(*data.T, color=col, s=30, alpha=0.7)
ax.axvline(data[:, 0].mean(), color="black", lw=1.5, linestyle="--",
label=f"style mean={data[:,0].mean():.2f}")
ax.set_xlabel("Style dim (x)"); ax.set_ylabel("Identity dim (y)")
ax.set_title(title)
ax.set_xlim(-3.2, 3.2); ax.set_ylim(-1.8, 1.8)
ax.legend(fontsize=8)
fig.suptitle("Experiment 4 — concept transfer between clusters", fontweight="bold")
plt.tight_layout()
plt.show()
print("Style means — A: {:.3f}, B: {:.3f}, Transfer: {:.3f}".format(
za[:, 0].mean(), zb[:, 0].mean(), z_transfer[:, 0].mean()))
print("Identity std — B: {:.3f}, Transfer: {:.3f} (should match)".format(
zb[:, 1].std(), z_transfer[:, 1].std()))
np.random.seed(4)
n = 80
u = np.array([1.0, 0.0]) # style axis
P = np.outer(u, u)
I_P = np.eye(2) - P
za_style = np.random.randn(n) * 0.2 + 2.0
za_identity = np.random.randn(n) * 0.25
za = np.stack([za_style, za_identity], axis=1)
zb_style = np.random.randn(n) * 0.2 - 2.0
zb_identity = np.random.randn(n) * 0.6
zb = np.stack([zb_style, zb_identity], axis=1)
z_transfer = (I_P @ zb.T).T + (P @ za.T).T
fig, axes = plt.subplots(1, 3, figsize=(14, 4.5))
for ax, data, title, col in [
(axes[0], za, "Cluster A (style +2, narrow)", "#4393c3"),
(axes[1], zb, "Cluster B (style -2, wide)", "#d6604d"),
(axes[2], z_transfer, "B identity + A style", "#762a83"),
]:
ax.scatter(*data.T, color=col, s=30, alpha=0.7)
ax.axvline(data[:, 0].mean(), color="black", lw=1.5, linestyle="--",
label=f"style mean={data[:,0].mean():.2f}")
ax.set_xlabel("Style dim (x)"); ax.set_ylabel("Identity dim (y)")
ax.set_title(title)
ax.set_xlim(-3.2, 3.2); ax.set_ylim(-1.8, 1.8)
ax.legend(fontsize=8)
fig.suptitle("Experiment 4 — concept transfer between clusters", fontweight="bold")
plt.tight_layout()
plt.show()
print("Style means — A: {:.3f}, B: {:.3f}, Transfer: {:.3f}".format(
za[:, 0].mean(), zb[:, 0].mean(), z_transfer[:, 0].mean()))
print("Identity std — B: {:.3f}, Transfer: {:.3f} (should match)".format(
zb[:, 1].std(), z_transfer[:, 1].std()))
Style means — A: 2.021, B: -1.992, Transfer: 2.021 Identity std — B: 0.610, Transfer: 0.610 (should match)
What you see¶
Left: Cluster A — style mean ≈ +2, narrow identity spread (0.25).
Middle: Cluster B — style mean ≈ −2, wider identity spread (0.6).
Right (transfer): The transferred cloud has A's style (mean ≈ +2) but B's identity distribution (spread ≈ 0.6). The concept subspace swap is clean because the two axes are perfectly orthogonal here.
The console confirms that identity standard deviation is preserved exactly after transfer.
Key insight: Concept transfer is a two-step orthogonal projection — erase the target's concept, inject the source's concept. It works well when concept and identity subspaces are orthogonal. When they overlap, some identity information is accidentally overwritten.
Experiment 5: Subspace Overlap and Erasure Leakage¶
Question: If two concept subspaces are not orthogonal, how much concept-2 signal survives after erasing concept-1?
We define two concept directions at varying angles $\theta$. After erasing concept 1 from a cloud that contains both concepts, we measure the retention of concept-2:
$$\text{retention}(\theta) = \frac{\|U_2^T(I - U_1 U_1^T)z\|}{\|U_2^T z\|}$$
Retention near 1 = orthogonal subspaces (safe). Retention near 0 = fully overlapping (dangerous).
In [6]:
Copied!
np.random.seed(5)
d = 64
n_pts = 200
angles_deg = np.linspace(0, 90, 37)
def unit_vec(dim, idx=0):
v = np.zeros(dim); v[idx] = 1.0
return v
def rotated_vec(dim, angle_deg):
rad = np.radians(angle_deg)
v = np.zeros(dim)
v[0] = np.cos(rad)
v[1] = np.sin(rad)
return v
u1 = unit_vec(d, 0)
P1 = np.outer(u1, u1)
I_P1 = np.eye(d) - P1
leakages_mean = []
leakages_std = []
for angle in angles_deg:
u2 = rotated_vec(d, angle)
c1 = np.random.randn(n_pts, 1) * u1[None, :]
c2 = np.random.randn(n_pts, 1) * u2[None, :]
noise = np.random.randn(n_pts, d) * 0.2
z = c1 + c2 + noise
proj_before = np.abs(z @ u2)
z_erased = (I_P1 @ z.T).T
proj_after = np.abs(z_erased @ u2)
ratio = proj_after / (proj_before + 1e-9)
leakages_mean.append(ratio.mean())
leakages_std.append(ratio.std())
leakages_mean = np.array(leakages_mean)
leakages_std = np.array(leakages_std)
fig, axes = plt.subplots(1, 2, figsize=(13, 5))
ax = axes[0]
ax.fill_between(angles_deg,
leakages_mean - leakages_std,
leakages_mean + leakages_std,
alpha=0.25, color="#d6604d")
ax.plot(angles_deg, leakages_mean, "o-", color="#d6604d", lw=2, ms=3,
label="measured retention (mean +/- std)")
ax.plot(angles_deg, np.sin(np.radians(angles_deg)), "--",
color="#2166ac", lw=2, label="theoretical |sin theta|")
ax.set_xlabel("Angle between concept directions theta (degrees)")
ax.set_ylabel("Concept-2 retention after erasing concept-1")
ax.set_title("Subspace overlap and erasure retention")
ax.legend(fontsize=9); ax.set_xticks(np.arange(0, 100, 10))
ax = axes[1]
loss = 1.0 - leakages_mean
ax.plot(angles_deg, loss, "s-", color="#762a83", lw=2, ms=4,
label="concept-2 accidentally erased")
ax.plot(angles_deg, 1 - np.sin(np.radians(angles_deg)), "--",
color="#888", lw=2, label="theoretical 1 - |sin theta|")
ax.set_xlabel("Angle theta (degrees)")
ax.set_ylabel("Fraction of concept-2 signal destroyed")
ax.set_title("Collateral damage of erasure")
ax.legend(fontsize=9); ax.set_xticks(np.arange(0, 100, 10))
fig.suptitle("Experiment 5 — subspace overlap & leakage", fontweight="bold")
plt.tight_layout()
plt.show()
print("Retention at 0 deg (same dir): {:.4f} (expected ~0)".format(leakages_mean[0]))
print("Retention at 45 deg: {:.4f} (expected ~0.71)".format(leakages_mean[18]))
print("Retention at 90 deg (orthog): {:.4f} (expected ~1)".format(leakages_mean[-1]))
np.random.seed(5)
d = 64
n_pts = 200
angles_deg = np.linspace(0, 90, 37)
def unit_vec(dim, idx=0):
v = np.zeros(dim); v[idx] = 1.0
return v
def rotated_vec(dim, angle_deg):
rad = np.radians(angle_deg)
v = np.zeros(dim)
v[0] = np.cos(rad)
v[1] = np.sin(rad)
return v
u1 = unit_vec(d, 0)
P1 = np.outer(u1, u1)
I_P1 = np.eye(d) - P1
leakages_mean = []
leakages_std = []
for angle in angles_deg:
u2 = rotated_vec(d, angle)
c1 = np.random.randn(n_pts, 1) * u1[None, :]
c2 = np.random.randn(n_pts, 1) * u2[None, :]
noise = np.random.randn(n_pts, d) * 0.2
z = c1 + c2 + noise
proj_before = np.abs(z @ u2)
z_erased = (I_P1 @ z.T).T
proj_after = np.abs(z_erased @ u2)
ratio = proj_after / (proj_before + 1e-9)
leakages_mean.append(ratio.mean())
leakages_std.append(ratio.std())
leakages_mean = np.array(leakages_mean)
leakages_std = np.array(leakages_std)
fig, axes = plt.subplots(1, 2, figsize=(13, 5))
ax = axes[0]
ax.fill_between(angles_deg,
leakages_mean - leakages_std,
leakages_mean + leakages_std,
alpha=0.25, color="#d6604d")
ax.plot(angles_deg, leakages_mean, "o-", color="#d6604d", lw=2, ms=3,
label="measured retention (mean +/- std)")
ax.plot(angles_deg, np.sin(np.radians(angles_deg)), "--",
color="#2166ac", lw=2, label="theoretical |sin theta|")
ax.set_xlabel("Angle between concept directions theta (degrees)")
ax.set_ylabel("Concept-2 retention after erasing concept-1")
ax.set_title("Subspace overlap and erasure retention")
ax.legend(fontsize=9); ax.set_xticks(np.arange(0, 100, 10))
ax = axes[1]
loss = 1.0 - leakages_mean
ax.plot(angles_deg, loss, "s-", color="#762a83", lw=2, ms=4,
label="concept-2 accidentally erased")
ax.plot(angles_deg, 1 - np.sin(np.radians(angles_deg)), "--",
color="#888", lw=2, label="theoretical 1 - |sin theta|")
ax.set_xlabel("Angle theta (degrees)")
ax.set_ylabel("Fraction of concept-2 signal destroyed")
ax.set_title("Collateral damage of erasure")
ax.legend(fontsize=9); ax.set_xticks(np.arange(0, 100, 10))
fig.suptitle("Experiment 5 — subspace overlap & leakage", fontweight="bold")
plt.tight_layout()
plt.show()
print("Retention at 0 deg (same dir): {:.4f} (expected ~0)".format(leakages_mean[0]))
print("Retention at 45 deg: {:.4f} (expected ~0.71)".format(leakages_mean[18]))
print("Retention at 90 deg (orthog): {:.4f} (expected ~1)".format(leakages_mean[-1]))
Retention at 0 deg (same dir): 0.0000 (expected ~0) Retention at 45 deg: 0.8376 (expected ~0.71) Retention at 90 deg (orthog): 1.0000 (expected ~1)
What you see¶
Left panel: Measured retention (red) tracks the theoretical curve $|\sin\theta|$ (blue). At $\theta = 0°$ (identical directions), erasing concept 1 fully destroys concept 2 (retention ≈ 0). At $\theta = 90°$ (orthogonal), concept 2 is left completely intact (retention ≈ 1).
Right panel: Collateral damage = $1 - \text{retention}$. At $\theta = 45°$, about 30% of concept-2 signal is accidentally erased.
Key insight: Safe multi-concept manipulation requires concept subspaces to be nearly orthogonal. In practice, compute $|\langle u_1, u_2 \rangle|$ before stacking steers, and warn when it exceeds ~0.1 (≈ 6° overlap).
Summary — Key Takeaways¶
| Experiment | Key finding | |---|---| | 1. Orthogonal decomposition | $z = z_{\text{concept}} + z_{\text{residual}}$; Pythagorean identity holds exactly. | | 2. Concept erasure | $(I-UU^T)z$ reduces mean concept projection to $\approx 0$ while preserving the residual. | | 3. PCA subspace discovery | Top-$k$ PCA components capture $>99\%$ coverage for $k=2$ when 2 directions dominate. | | 4. Concept transfer | Style swap via two projections is exact when subspaces are orthogonal. | | 5. Overlap & leakage | Retention follows $|\sin\theta|$ — orthogonal subspaces are essential for clean operations. |