Cognitive　functions　are　often　studied　using　event-related　potentials　(ERPs)　that　are　usually　estimated　by　an　averaging　algorithm.　Clearly,　estimation　of　single-trial　ERPs　can　provide　researchers　with　many　more　details　of　cognitive　activity　than　the　averaging　algorithm.　A　novel　method　to　estimate　single-trial　ERPs　is　proposed　in　this　paper.　This　method　includes　two　key　ideas.　First,　singular　value　decomposition　was　used　to　construct　a　matrix,　which　mapped　single-trial　electroencephalographic　recordings　(EEG)　into　a　low-dimensional　vector　that　contained　little　information　from　the　spontaneous　EEG.　Second,　we　used　the　theory　of　compressed　sensing　to　build　a　procedure　to　restore　single-trial　ERPs　from　this　low-dimensional　vector.　ERPs　are　sparse　or　approximately　sparse　in　the　frequency　domain.　This　fact　allowed　us　to　use　the　theory　of　compressed　sensing.　We　verified　this　method　in　simulated　and　real　data.　Our　method　and　dVCA　(differentially　variable　component　analysis),　another　method　of　single-trial　ERPs　estimation,　were　both　used　to　estimate　single-trial　ERPs　from　the　same　simulated　data.　Results　demonstrated　that　our　method　significantly　outperforms　dVCA　under　various　conditions　of　signal-to-noise　ratio.　Moreover,　the　single-trial　ERPs　estimated　from　the　real　data　by　our　method　are　statistically　consistent　with　the　theories　of　cognitive　science.　©　2013　Shanghai　Institutes　for　Biological　Sciences,　CAS　and　Springer-Verlag　Berlin　Heidelberg.