Background Gradients Shape Memorization in Flow Matching

Xuanhua Yin; Boyu Wei; Shuyi Zhang; Shunqi Mao; Chuanzhi Xu; Weidong Cai

cs.CVSep 27, 2026

Background Gradients Shape Memorization in Flow Matching

Authors: Xuanhua Yin, Boyu Wei, Shuyi Zhang, Shunqi Mao, Chuanzhi Xu, Weidong Cai

Organizations: School of Computer Science, The University of Sydney · Institute of Automation, Chinese Academy of Sciences

Abstract

Repetition is closely associated with memorization in generative models, but how other training images affect the retention and copying of targets remains unclear. We study this question in class-conditioned flow matching, where images outside the target set form the background. At fixed target repetition and same-class background row count, replacing repeated same-class images with distinct images reduces the target extraction rate from 80.7% to 18.0%. To explain this effect, we develop a paired-trajectory framework that isolates target-induced parameter displacement and the background gradient response to it. This response has an exact path-integrated curvature representation, connecting background loss geometry to target learning. Reciprocal response transfer between repeated and distinct backgrounds changes target retention and copying in both directions, establishing the response's causal role. After target removal, the response correction parallel to the target-induced displacement preserves approximately 90% of the copying effects of full response transfer. Directly scaling the displacement also changes copying without further training. The post-removal copying effects of reciprocal transfer are reproduced across datasets and architectures. Together, these results identify the background gradient response as a mechanism through which same-class training data shape the retention of target learning and the reproduction of target images.

Figures & tables

Figure 1: Data composition and target learning. Left: at 400 same-class background rows, repeated ( $R$ ) and distinct ( $U$ ) content give 80.7% and 18.0% target extraction, pooled over target repetition counts. Pictured target copies are schematic. Middle: balanced-design target extraction. Right: reciprocal response transfer and interleaved per-generation target copying.

Figure 2: Paired training and reciprocal response transfer. Target-on and target-off states define the target-induced parameter displacement. At the current recipient pair, an alternative background supplies the response while the recipient retains its baseline background gradient. Continuing training measures the resulting target retention and copying.

Figure 3: Distinct background content reduces target extraction, and reciprocal response transfer changes target-specific benefit and copying. (a) Extraction under matched row or distinct-image counts. (b) Corresponding conditional accuracy. (c) Effective repetition of photometric variants relative to exact copies under three detection-reference policies. (d) Finite-response projection changes (%) and normalized-benefit differences from native values. (e) Copying contrasts (transfer minus comparator, percentage points), with paired 95% confidence intervals across eight independent panels. In (d) and (e), orange denotes $R\leftarrow U$ and blue denotes $U\leftarrow R$ after target removal or during interleaved training, where $R$ and $U$ use repeated and distinct backgrounds, respectively.

Background	$h_{S}$	$\mathcal{R}$	Target copying rate (%)
$R$ , $25\times 16$	0.03834	0.3206	2.520 [2.197, 2.843]
$M$ , $100\times 4$	0.05200	0.2197	1.177 [0.978, 1.376]
$U$ , $400\times 1$	0.06499	0.1305	0.439 [0.372, 0.506]

Table 1: Native backgrounds across eight experimental panels. Target-direction curvature

h_{S}

uses the shared step-2,500 checkpoint. Normalized target-specific benefit and target copying rate use interleaved training at step 6,000. Brackets give 95% panel intervals.

	$R\leftarrow U$		$U\leftarrow R$
Correction	Copying	Effect [95% CI]	Copying	Effect [95% CI]
Native	2.685	n/a	0.523	n/a
Full	1.732	0.953 [0.912, 0.994]	1.622	1.099 [1.042, 1.156]
Parallel only	1.820	0.865 [0.832, 0.898]	1.500	0.977 [0.928, 1.026]
Orthogonal only	2.570	0.115 [0.107, 0.123]	0.640	0.117 [0.109, 0.125]

Table 2: Response-component interventions after target removal. Copying rates are percentages. Effects are native minus intervention for

R

and intervention minus native for

U

, in percentage points. Brackets are ordinary 95% paired

t

intervals from the eight-panel component experiment.

State or direction	Target copy	$B^{\mathrm{spec}}$	Decoy hit	Non-target copy
$\gamma=0$	0.33	0	0.30	0.45
$\gamma=0.5$	1.30	0.071	0.30	0.45
$\gamma=1$	2.520	0.1284	0.30	0.46
$\gamma=1.5$	3.71	0.172	0.31	0.47
Random parameter direction	0.38	0.004	0.30	0.45
Non-target learning direction	0.44	0.006	0.31	0.46

Table 3: State interventions on eight native interleaved

R

pairs at step 6,000. Entries are panel means, and rates are percentages.

B^{\mathrm{spec}}

uses a common target-off reference. The random direction matches the

\gamma=1

displacement norm.

Setting	Comparator	$R\leftarrow U$		$U\leftarrow R$
Setting	Comparator	Change	95% CI	Change	95% CI
CIFAR-10 / DiT	Native	$\mathbf{-0.953}$	$[-1.277,-0.630]$	$\mathbf{+1.099}$	$[0.929,1.269]$
CIFAR-10 / DiT	Random	$\mathbf{-0.676}$	$[-0.863,-0.489]$	$\mathbf{+0.986}$	$[0.805,1.167]$
CINIC-10 / DiT	Native	$\mathbf{-0.682}$	$[-0.778,-0.586]$	$\mathbf{+0.711}$	$[0.558,0.864]$
CINIC-10 / DiT	Random	$\mathbf{-0.584}$	$[-0.716,-0.453]$	$\mathbf{+0.615}$	$[0.485,0.744]$
CIFAR-10 / U-Net	Native	$\mathbf{-0.568}$	$[-0.722,-0.414]$	$\mathbf{+0.568}$	$[0.494,0.642]$

Table 4: Post-removal changes in target copying (percentage points). Values are transfer minus comparator, with paired 95% confidence intervals across panels (

n=8

for CIFAR-10/DiT and

n=6

otherwise). Random denotes the matched random-direction control.

Appendix figures & tables37 assets

Supplementary material from the paper’s appendix.

Appendix

Regime	Condition	$B^{\mathrm{spec}}$	$\mathcal{R}$
Interleaved	R native	0.1284	0.3206
	M native	0.0879	0.2197
	U native	0.0522	0.1305
	R receiving U response	0.0784	0.1959
	U receiving R response	0.0987	0.2475
Pulse-chase	R native	0.1412	0.3517

Appendix

Table 5: Signed target-specific benefit and its normalized counterpart at step 6,000. The native and transferred arms use eight paired panels. Each normalizer is fixed before continuation and shared across conditions for the same panel-target. All four transfer comparisons agree in direction between

B^{\mathrm{spec}}

and

\mathcal{R}

in all eight panels.

Background	LPIPS copy	Accuracy	KID
R	3.403	71.299	0.04121
M	1.595	70.636	0.03899
U	0.596	69.719	0.03741

Appendix

Table 6: Native interleaved generation quality and LPIPS copying at step 6,000. Rates are percentages. KID uses unscaled Inception pool3 features. The R row also describes the shared

\gamma=1

dose endpoint.

Receiver	Intervention	$\mathcal{R}$	Pixel copy [95% CI]	$\Delta$ Acc.	$\Delta$ KID
Pulse-chase
R	Native	0.3517	2.685 [2.388, 2.981]	n/a	n/a
R	U response	0.1954	1.732 [1.502, 1.961]	$-0.341$	$-0.00128$
R	Matched random	0.3146	2.408 [2.220, 2.596]	$-0.068$	$-0.00054$
R	Preclip norm	0.3157	2.424 [2.105, 2.743]	$-0.063$	$-0.00038$
U	Native	0.1428	0.523 [0.446, 0.600]	n/a	n/a

Appendix

Table 7: Reciprocal response transfer and controls with full response replacement and eight paired panels per regime. The receiver supplies the training background, and the source supplies the response. Copy rates are percentages with 95% panel intervals. Accuracy changes (pp) and KID changes are relative to the same receiver’s native condition. The n/a entries mark native reference rows, for which changes are not reported. Random and norm controls use the corresponding transfer’s matching constraints.

Receiver	Intervention	Pulse-chase	Interleaved
R	U response	$+67.3$ [ $+65.5$ , $+69.0$ ]	$+68.0$ [ $+65.3$ , $+70.7$ ]
U	R response	$-39.4$ [ $-41.2$ , $-37.6$ ]	$-39.7$ [ $-40.9$ , $-38.5$ ]

Appendix

Table 8: Finite-response projection changes (%) under reciprocal transfer. Source and native projections are measured at matched recipient states, then averaged over four times and four batches per time within each panel before taking their ratio (Appendix A.1 ). Entries give means and 95% intervals across eight panels. Shared-checkpoint curvature

h_{S}

is reported separately in Table 1 .

Phase	Step interval	Updates	Target draws
Common target learning	2,500–2,900	400	640
Interleaved continuation	2,900–6,000	3,100	4,960
Pulse-chase continuation	2,900–6,000	3,100	0

Appendix

Table 9: Training schedule and actual target exposure. Counts are per target in the target-on branch, whereas target-off branches omit the target loss.

Diagnostic	$R\leftarrow U$	$U\leftarrow R$
$\alpha_{s}$ , median [IQR]	1.08 [1.03, 1.14]	0.93 [0.89, 0.97]
Maximum $\lVert G_{\mathrm{norm}}\rVert_{2}/\lVert G_{\mathrm{tr}}\rVert_{2}-1$	$3.1\times 10^{-7}$	$2.8\times 10^{-7}$
Maximum $\lVert r_{\ell}\rVert_{2}/\lVert\Delta G_{\ell}\rVert_{2}-1$	$4.4\times 10^{-7}$	$4.0\times 10^{-7}$
Zero-norm steps	0	0
Zero-correction groups per step	0	0

Appendix

Table 10: Gradient-control diagnostics for reciprocal pulse-chase transfers. Norm-ratio deviations are measured before clipping. IQR denotes the interquartile range.

Experiment	Evaluation point	Images	Endpoint
Balanced	Step 10,000, EMA	2,000	Target extraction rate
Composition	Step 10,000	2,000	Target extraction rate
Mechanism	Step 6,000, raw parameters	4,096	Target copying rate

Appendix

Table 11: Evaluation settings. Generation budgets are per class and model. Target extraction rate is an any-hit endpoint over target images, and target copying rate is a per-generation endpoint. Photometric evaluation follows Appendix G.

Regime	Receiver / comparator	Effect [95% CI]	Holm $p$
Pulse-chase	R / native	0.953 [0.630, 1.277]	$2.17\times 10^{-4}$
	U / native	1.099 [0.929, 1.269]	$4.97\times 10^{-6}$
	R / random	0.676 [0.489, 0.863]	$1.21\times 10^{-4}$
	U / random	0.986 [0.805, 1.167]	$1.17\times 10^{-5}$
	R / preclip	0.692 [0.486, 0.898]	n/a
	U / preclip	0.948 [0.758, 1.137]	n/a

Appendix

Table 12: Paired copying contrasts across eight panels in each primary regime. The four native/random comparisons constitute one Holm family per regime. Preclip comparisons are supplementary and are not assigned a

p

value from that four-test family (n/a).

Receiver / comparator	Pulse-chase	Interleaved
R / native	0.1562 [0.1298, 0.1827]	0.1247 [0.1072, 0.1422]
U / native	0.1494 [0.1413, 0.1575]	0.1170 [0.0993, 0.1348]
R / random	0.1191 [0.1005, 0.1378]	0.1123 [0.0969, 0.1276]
U / random	0.1383 [0.1264, 0.1503]	0.1089 [0.0933, 0.1245]
R / preclip	0.1203 [0.1005, 0.1401]	0.0984 [0.0742, 0.1226]
U / preclip	0.1253 [0.1074, 0.1433]	0.1050 [0.0859, 0.1242]

Appendix

Table 13: Paired differences in normalized target-specific benefit, with ordinary 95% panel intervals. Each entry uses eight panels and the target-wise normalization in Equation 11 . The signs follow the copying contrasts: comparator minus transfer for R, and transfer minus comparator for U.

CINIC-10 / DiT
Receiver	Native	Transfer	Random	Beyond random
R	2.000 [1.606, 2.394]	1.318 [1.005, 1.630]	1.902 [1.516, 2.288]	0.584 [0.453, 0.716]
U	0.446 [0.347, 0.545]	1.157 [0.924, 1.389]	0.542 [0.426, 0.658]	0.615 [0.485, 0.744]
CIFAR-10 / U-Net FM
R	1.742 [1.538, 1.947]	1.174 [1.086, 1.262]	1.662 [1.492, 1.832]	0.488 [0.386, 0.590]
U	0.401 [0.374, 0.428]	0.969 [0.897, 1.041]	0.466 [0.408, 0.524]	0.503 [0.431, 0.574]

Appendix

Table 14: Complete pulse-chase rates in the additional settings, each with six panels. Native, transfer, and random columns give copying percentages with 95% panel intervals. The last column gives the paired beyond-random effect in percentage points. Settings are analyzed separately.

Setting	Experimental panels	Native $R$ minus $U$
CIFAR-10 with DiT, primary	8	2.162 [1.884, 2.440]
CINIC-10 with DiT	6	1.554 [1.249, 1.859]
CIFAR-10 with U-Net	6	1.341 [1.143, 1.540]

Appendix

Table 15: Native background gaps during continued training after target removal. Entries give the native

R

minus

U

target copying rate in percentage points, with paired 95% panel intervals. The primary setting is included as the reference comparison.

Background composition	Target extraction rate	Conditional accuracy
Same-class $400\times 1$	18.0	56.6
Same-class $25\times 16$	80.7	93.9
Same-class $25\times 1$	86.0	64.9
Relabeled other-class $400\times 1$	20.0	8.3
Relabeled other-class $25\times 16$	82.0	5.8

Appendix

Table 16: Composition means over 150 paired target evaluations per condition. Entries specify distinct images multiplied by repetition counts. Both rates are percentages. Relabeled other-class images retain the target conditioning label.

Figure 4: Same-class data composition changes target extraction rate. (a) The balanced design contains 30 target images per cell and 2,000 generated images per class and model. Cell labels are rounded percentages, and the outline marks $k=4$ . (b) Composition means over 150 paired target evaluations per condition. Labels give distinct images multiplied by their repetition counts. The shaded conditions contain 400 other same-class training examples. Full coefficients and block analyses appear in Appendix H.

Figure 5: Reciprocal response transfer across settings during continued training after target removal. Changes in target copying rate are measured against (a) native training and (b) random-direction controls. Negative values indicate reduced copying under $R\leftarrow U$ , and positive values indicate increased copying under $U\leftarrow R$ . Intervals are paired 95% panel intervals. The primary CIFAR-10 with DiT result provides the reference comparison with eight experimental panels. Each additional setting uses six experimental panels. Native background gaps appear in Table 15 .

Figure 6: Displacement scaling connects retained target learning to generated copying. (a) Target copying rate across scaling factors for the step-6,000 native interleaved $R$ pair. (b) Target-specific benefit across the same scaling factors, using the same target-off reference and loss evaluation bank. (c) The $\gamma=1$ target-induced parameter displacement compared with a random parameter-direction control and a non-target learning direction. The random control matches the $\gamma=1$ displacement norm. (d) Decoy hit rate and non-target copying rate across the same scaling factors. Points are reported means over eight experimental panels. Lines connect evaluated scaling factors, and all conditions share generation inputs and detection settings.

Figure 7: Controlled penalty orientation and short-horizon target learning. Runs 1 and 2 are independent models of the same architecture. Primary and independent evaluation banks reuse the same ten targets with different noise. Bars show low-minus-high learning effects with 95% paired hierarchical-bootstrap intervals. Percentages show relative reductions. The orientation panel is schematic. Predictions and observations use units of $10^{-7}$ mean FM loss.

Symbol	Meaning
$x,c,\epsilon,t$	Image, conditioning class, Gaussian noise, and flow time.
$\theta,v_{\theta}$	Network parameters and class-conditioned velocity field.
$k,N_{\mathrm{bg}}$	Target-copy count and distinct non-target same-class image count in the balanced design.
$b,r$	Cycle and training-run indices.
$A_{c}(y),E_{bc}^{S}$	Attributable hit for one generated image and at least one hit for a target in a model, respectively.
$a,b_{N},\lambda$	Repetition coefficient, support coefficient, and fitted exponent $-b_{N}/a$ .

Appendix

Table 17: Additional notation used in the earlier data and controlled-geometry experiments. Flow time describes the noise-to-data interpolation. The update index describes optimization.

Data experiment	Untrained-reference extraction (%)
Balanced	3.0
Composition	2.7
Near-duplicate	3.3

Appendix

Table 18: Untrained-reference extraction under the matched detector and generation budget. Each entry is an any-hit rate over

M=2{,}000

generations.

Reference policy	Light variations	Strong variations
Original image	0.987 [0.968, 1.006]	0.983 [0.963, 1.005]
One training variant	0.996 [0.978, 1.014]	0.982 [0.959, 1.006]

Appendix

Table 19: Effective repetition relative to exact copies under the two one-reference policies. Brackets are 90% paired target-cluster bootstrap intervals conditional on the five paired training seeds.

Figure 8: Complete balanced-design results: pixel and LPIPS extraction, endpoint-specific support exponents with 95% Fieller intervals, and held-cycle prediction. The gray exponent band is the prespecified $[0.4,0.6]$ reference range, not an uncertainty interval. Each heatmap cell contains 30 targets.

Endpoint	Quantity	Estimate	95% interval
Pixel $\ell_{2}$	Repetition coefficient $a$	3.498	[3.098, 3.898]
	Support coefficient $b_{N}$	$-2.297$	[ $-2.607$ , $-1.987$ ]
	Support exponent $\lambda$	0.657	[0.610, 0.703]
LPIPS-Alex	Support exponent $\lambda$	1.043	[0.912, 1.181]

Appendix

Table 20: Balanced extraction estimates at the 10,000-step EMA checkpoint. Coefficient intervals use two-way target/run cluster-robust covariance. Exponent intervals use Fieller’s method. Pixel fitting uses 725 informative observations, and LPIPS evaluates the same generated pools.

Background contrast	Extraction change	Log-odds [95% interval]	Accuracy change
Same-class: 400 distinct vs. $25\times 16$	$-62.67$	$-6.76$ $[-9.86,-4.78]$	$-37.29$
Other-class: 400 distinct vs. $25\times 16$	$-62.00$	$-5.83$ $[-8.42,-4.23]$	$+2.49$
Same-class: $25\times 16$ vs. 25 distinct	$-5.33$	$-1.68$ $[-3.29,-0.43]$	$+29.03$
400 distinct: same-class vs. other-class	$-2.00$	$-1.20$ $[-2.89,0.34]$	$+48.37$

Appendix

Table 21: Composition results for 150 paired targets across three blocks. Contrasts are left minus right. Log-odds estimates and 95% profile intervals come from the post hoc conditional Firth analysis. Extraction and accuracy differences are percentage points. Accuracy uses the intended conditioning class.

Background contrast	Block 1	Block 2	Block 3	Mean	95% interval
Same-class: 400 distinct vs. $25\times 16$	$-60.9$	$-63.1$	$-64.0$	$-62.6667$	$[-66.6283,-58.7050]$
Other-class: 400 distinct vs. $25\times 16$	$-60.2$	$-62.5$	$-63.3$	$-62.0000$	$[-65.9978,-58.0022]$
Same-class: $25\times 16$ vs. 25 distinct	$-3.8$	$-5.9$	$-6.3$	$-5.3333$	$[-8.6692,-1.9974]$

Appendix

Table 22: Composition extraction contrasts by training block, in percentage points. Contrasts have the same orientation as Table 21 . The last column gives ordinary, unadjusted 95%

t

intervals across the three blocks. All three leave-one-block-out estimates retain the pooled orientation.

Condition	Block 1	Block 2	Block 3
U	0 / 44.02	3 / 45.36	0 / 46.25
R	268 / 71.13	357 / 71.49	531 / 70.90
U $\leftarrow$ R	0 / 44.26	3 / 45.64	0 / 44.95
R $\leftarrow$ U	93 / 68.59	123 / 69.35	142 / 68.91
U + orthogonal control	0 / 42.22	2 / 44.49	0 / 45.21
R + orthogonal control	106 / 55.25	124 / 43.30	130 / 50.44

Appendix

Table 23: All native-continuation blocks. Each entry gives attributable copy count / conditional accuracy (%), out of 20,000 generated images. U denotes distinct backgrounds. R denotes repeated backgrounds. Arrows denote the donor of the exchanged component.

Model	Bank	Improvement difference	Relative reduction	95% lower bound
1	H	3.547	17.83%	1.629
1	D	3.722	18.14%	1.653
2	H	3.390	15.47%	1.890
2	D	3.542	15.73%	2.061

Appendix

Table 24: Target-loss improvement after eight SGD steps. Differences are low minus high alignment, in

10^{-7}

mean FM-loss units. H and D are evaluation noise banks. Each initial model uses the same ten targets in both banks. Uncertainty is clustered by target.

Model	Cross-target slope	Pearson $r$	LOTO $Q^{2}$
1	0.9950	0.999979	0.999894
2	0.9978	0.999863	0.999623

Appendix

Table 25: Full-network predictions, separately by initial model. The high-minus-low final loss difference equals the low-minus-high improvement difference. Leave-one-target-out (LOTO) calibration fits nine targets and predicts the tenth. The recurrence is computed before accessing post-intervention losses.

Figure 9: Composition and near-duplicate results. (a) Pooled target extraction for 150 paired targets per condition. (b) Conditioning-class accuracy from 30 block-class observations, with diamonds for pooled means and faint points for block means. Distinct-content comparisons hold background row counts fixed. Other-class images are relabeled. (c) Effective repetition ratios with 90% paired target-cluster bootstrap intervals under the three reference policies. The gray band marks the equivalence range $[0.8,1.25]$ .

Figure 10: Near-duplicate reference controls. Points are effective repetition ratios and bars are 90% paired target-cluster bootstrap intervals conditional on the five paired training seeds. The gray band is the prespecified equivalence range $[0.8,1.25]$ . The design includes five paired seeds, three training arms, and five checkpoints, with 750 seed-target clusters preserving checkpoint and reference-policy pairing.

	Added inputs	Retention MAE	Retention RMSE	Open-loop error	Copy log-loss	Copy Brier
P0	Baseline inputs	0.1427	0.1846	0.2241	0.4131	0.1361
P1	First-order gradients	0.1220	0.1577	0.1952	0.3722	0.1186
P2	Global curvature, update scale	0.1163	0.1504	0.1831	0.3578	0.1133
P3	Target-relevant response	0.0950	0.1228	0.1502	0.3114	0.0964
P4	AdamW state dynamics	0.0932	0.1205	0.1468	0.3049	0.0941

Appendix

Table 26: Eight-fold leave-one-panel-out prediction across 80 panel-target observations, with 70 training and ten test observations per fold. P0 uses exposure, initial losses, and quality. Successive rows add the named inputs, all measured at step 2,900. Open-loop error is evaluated over 256 steps and normalized by the initial target-specific benefit. Lower values are better.

Background	Training steps	Extraction	Accuracy
$25\times 16$	10,000	80.67	93.92
$400\times 1$	10,000	18.00	56.63
$25\times 1$	10,000	86.00	64.89
$25\times 16$	20,000	78.00	94.00
$400\times 1$	20,000	30.00	80.00

Appendix

Table 27: Same-class composition at two training budgets. Extraction and conditional accuracy are percentages. Both same-class 400-row conditions are evaluated at each budget. These are equal-training-step comparisons, with accuracy reported as an outcome. The

25\times 1

arm is reported at 10,000 steps.

Step	R interleaved	R pulse-chase	U interleaved	U pulse-chase
Signed target-specific benefit
2,900	0.4010	0.4010	0.4010	0.4010
2,901	0.3950	0.3970	0.3920	0.3940
2,910	0.3600	0.3700	0.3500	0.3600
2,950	0.3000	0.3200	0.2800	0.2900
3,000	0.2500	0.2800	0.2100	0.2300

Appendix

Table 28: Signed target-specific benefit

B^{\mathrm{spec}}

and target-on FM loss along native continuations. Columns refer to background and training regime. All four continuations share the step-2,900 start.

Background	Regime	Target off	Target on	Decoy off	Decoy on	$B^{\mathrm{spec}}$	$\mathcal{R}$	Copy
R	Interleaved	0.5000	0.3516	0.4200	0.4000	0.1284	0.3206	2.520
R	Pulse-chase	0.5000	0.3388	0.4200	0.4000	0.1412	0.3517	2.685
U	Interleaved	0.5000	0.4278	0.4200	0.4000	0.0522	0.1305	0.439
U	Pulse-chase	0.5000	0.4229	0.4200	0.4000	0.0571	0.1428	0.523

Appendix

Table 29: The four losses defining signed target-specific benefit at step 6,000, shown to four decimal places. Off and on identify the paired branches. Target and decoy images share evaluation times and noise. Each loss is evaluated at its own branch parameters. Copying is a percentage.

Receiver	Condition	$B^{\mathrm{spec}}$	$\mathcal{R}$	Copy
R	Native	0.1412	0.3517	2.685
R	Earlier projection exchange	0.1100	0.2750	2.200
R	Full U response	0.0782	0.1954	1.732
R	Projection-exchange control	0.1380	0.3440	2.620
R	Full-response control	0.1360	0.3390	2.600
U	Native	0.0571	0.1428	0.523

Appendix

Table 30: Means for the same-protocol comparison of projection exchange and full response transfer, each with its operation-specific control. Copying is a percentage.

Receiver	Arm	$B^{\mathrm{spec}}$	$\mathcal{R}$	Copy	Accuracy	KID
R	Native	0.1412	0.3517	2.685	90.000	0.01000
R	Full response	0.0782	0.1954	1.732	89.659	0.00872
R	Along displacement only	0.0840	0.2100	1.820	89.800	0.00930
R	Orthogonal only	0.1320	0.3300	2.570	90.100	0.01010
U	Native	0.0571	0.1428	0.523	88.000	0.01200
U	Full response	0.1171	0.2922	1.622	88.688	0.01372

Appendix

Table 31: Response-component assay: means over eight paired panels. Copying and conditional accuracy are percentages. R receives components from U, and U receives components from R.

Receiver	Contrast	Mean	Paired SD	95% interval
R	Full-response effect	0.953	0.04899	[0.9120, 0.9940]
R	Along-displacement effect	0.865	0.03919	[0.8322, 0.8978]
R	Orthogonal effect	0.115	0.00980	[0.1068, 0.1232]
R	Along-displacement minus orthogonal effect	0.750	0.02939	[0.7254, 0.7746]
R	Along-displacement minus full effect	$-0.088$	0.00980	[ $-0.0962$ , $-0.0798$ ]
U	Full-response effect	1.099	0.06859	[1.0417, 1.1563]

Appendix

Table 32: Paired copying effects in the response-component assay, in percentage points. An arm’s effect is native minus arm in R and arm minus native in U, so positive effects follow the full transfer’s direction. Effect differences use the order shown. Intervals are ordinary, unadjusted 95%

t

intervals over eight paired panels.

	Receiver R				Receiver U
Panel	Native	Full	Random	Preclip	Native	Full	Random	Preclip
panel00	2.033691	1.889648	2.402344	2.910156	0.478516	1.682129	0.534668	0.544434
panel01	3.071289	2.209473	2.500000	2.949219	0.722656	2.067871	0.769043	0.854492
panel02	2.956543	1.928711	2.585449	2.470703	0.512695	1.682129	0.637207	0.764160
panel03	2.436523	1.511230	2.290039	2.399902	0.451660	1.257324	0.551758	0.573730
panel04	2.851562	1.838379	2.744141	2.353516	0.498047	1.853027	0.732422	0.637207

Appendix

Table 33: Primary pulse-chase panel inputs. Entries are copying percentages, shown to six decimal places. Calculations use the unrounded CSV values. Full transfers U response into R and R response into U. Random and Preclip denote the matched-random and preclip-norm controls.

	Receiver R				Receiver U
Panel	Native	Full	Parallel	Orthogonal	Native	Full	Parallel	Orthogonal
S01	2.475	1.592	1.666	2.374	0.453	1.454	1.346	0.556
S02	2.535	1.632	1.710	2.430	0.473	1.502	1.390	0.580
S03	2.595	1.672	1.754	2.486	0.493	1.550	1.434	0.604
S04	2.655	1.712	1.798	2.542	0.513	1.598	1.478	0.628
S05	2.715	1.752	1.842	2.598	0.533	1.646	1.522	0.652

Appendix

Table 34: Response-component panel inputs. Entries are copying percentages, retaining the supplied three-decimal precision. Parallel and Orthogonal refer to the correction components along and orthogonal to the recipient’s target-induced displacement. Full transfers U response into R and R response into U. The S01 to S08 labels identify panels within this record set.

Memorization in Generative Models

Memorization in generative models studies when models retain training examples, how to detect their recall, and how to limit it without losing utility.

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Aug 30, 2026

Tracing Generated Samples to Training-Data Clusters in Flow-Matching Models

Understanding which training samples influence a generated image is an important problem in generative modeling. In flow matching, training samples influence the generated image through the velocity field along the generation trajectory. Removing samples to examine their counterfactual influence changes the velocity field, and the resulting effect on the final image depends on how the change propagates through the trajectory. Consequently, local changes in the velocity field do not necessarily predict the final counterfactual effect. This work investigates attribution in flow-matching models through a hybrid analytical--learned approach, and uses it to derive trajectory-based attribution scores at the cluster level. We evaluate these attribution scores using independently retrained leave-one-cluster-out (LOCO) models, and compare with several attribution baselines using two different flow-matching latent spaces. Our experiments show that semantic similarity constitutes a strong baseline, while the closed-form trajectory-based attribution is competitive in some metrics without requiring counterfactual retraining or model gradients. Our results show that attribution in flow matching depends not only on semantic similarity to training samples, but also on the latent representation, trajectory dynamics, and how influence is propagated to the final output.

Rania Briq, Ohad Fried, Michael Kamp +1

Forschungszentrum Jülich · Technical University Dortmund · Reichman University +4

Flow Matching Flow-Based Generative Modeling Training Data Attribution

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Jun 9, 2026

Exploring the Design Space of Reward Backpropagation for Flow Matching

Aligning text-to-image flow matching models with human preferences via direct reward backpropagation is sample-efficient but hampered by two well-known pathologies: activations cannot be stored across the full sampling trajectory at modern model scale, and chained Jacobian products across steps inflate the reward gradient as it travels back to early indices. Connector-based methods, such as LeapAlign, address these issues by replacing the full backward trajectory with a short pinned path, highlighting a useful decoupling between sampling and optimization. However, the quality of the resulting gradient depends on how accurately this short path approximates the full rollout, especially over long intervals. We propose FlowBP, a unified surrogate-trajectory framework that treats the backward trajectory itself as the design object. FlowBP keeps a no-gradient cached rollout for sampling, then builds a lightweight backward surrogate from cached and selectively re-forwarded velocities. This view separates four choices: the reward-model input, active set, integration weights, and bridge coupling, and recovers prior direct-gradient methods as particular settings. Within this framework, we instantiate three variants: FlowBP-Sparse uses sparse Euler reconstruction, FlowBP-Bridge adds controlled bridge coupling, and FlowBP-Lagrange raises the order of leap quadrature. All three bound memory by the active-set size and limit gradient chaining to at most one Jacobian factor. Across SD3.5-M, FLUX.1-dev, and FLUX.2-Klein-base on preference, quality, and compositional metrics, the three variants improve over direct-gradient baselines on most metrics.

Ruoyu Wang, Boye Niu, Xiangxin Zhou +3

1Westlake University · 2Tencent Hunyuan · University of Sydney +1

Flow Matching AI Alignment T2I Generation Flow-Based Generative Modeling

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May 1, 2026

Posterior Augmented Flow Matching

Flow matching (FM) trains a time-dependent vector field that transports samples from a simple prior to a complex data distribution. However, for high-dimensional images, each training sample supervises only a single trajectory and intermediate point, yielding an extremely sparse and high-variance training signal. This under-constrained supervision can cause flow collapse, where the learned dynamics memorize specific source-target pairings, mapping diverse inputs to overly similar outputs, failing to generalize. We introduce Posterior-Augmented Flow Matching (PAFM), a theoretically grounded generalization of FM that replaces single-target supervision with an expectation over an approximate posterior of valid target completions for a given intermediate state and condition. PAFM factorizes this intractable posterior into (i) the likelihood of the intermediate under a hypothesized endpoint and (ii) the prior probability of that endpoint under the condition, and uses an importance sampling scheme to construct a mixture over multiple candidate targets. We prove that PAFM yields an unbiased estimator of the original FM objective while substantially reducing gradient variance during training by aggregating information from many plausible continuation trajectories per intermediate. Finally, we show that PAFM improves over FM by up to 3.4 FID50K across different model scales (SiT-B/2 and SiT-XL/2), different architectures (SiT and MMDiT), and in both class and text conditioned benchmarks (ImageNet and CC12M), with a negligible increase in the compute overhead. Code: https://github.com/gstoica27/PAFM.git.

George Stoica, Sayak Paul, Matthew Wallingford +6

Georgia Tech · University of Washington · Hugging Face +2

Flow Matching Conditional Image Generation Flow-Based Generative Modeling Variance Reduction

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Abstract

Memorization in Generative Models

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Tracing Generated Samples to Training-Data Clusters in Flow-Matching Models

Exploring the Design Space of Reward Backpropagation for Flow Matching

Posterior Augmented Flow Matching